On the distribution of the Picard ranks of the reductions of a $K3$ surface

We report on our results concerning the distribution of the geometric Picard ranks of $K3$ surfaces under reduction modulo various primes. In the situation that $\rk \Pic S_{\overline{K}}$ is even, we introduce a quadratic character, called the jump character, such that $\rk \Pic S_{\overline\bbF_{\!\frakp}}>\rk \Pic S_{\overline{K}}$ for all good primes, at which the character evaluates to $(-1)$. As an application, we obtain a sufficient criterion for the existence of infinitely many rational curves on a $K3$ surface of even geometric Picard rank. Our investigations also provide, as a by-product, a canonical choice of sign for the discriminant of an even-dimensional complete intersection.


Introduction
Let S be a K3 surface over a number field K. It is a well-known fact that the geometric Picard rank of S may not decrease under reduction modulo a good prime p of S. I.e., one always has rk Pic S p ě rk Pic S K .
(1) It would certainly be interesting to understand the sequence prk Pic S p q p , or at least the set of jump primes Π jump pSq :" t p prime of K | p good for S, rk Pic S p ą rk Pic S K u , for a given surface. In an ideal case, one would be able to give a precise reason why the geometric Picard rank jumps at a given good prime. There are two well-known such reasons. i) The Tate conjecture predicts that the left hand side is always even. Thus, in the case that rk Pic S K is odd, inequality (1) is always strict and every good prime is a jump prime. ii) Generalising this, if S has real multiplication by an endomorphism field E and the integer p22´rk Pic S K q{rE : És is odd then again every good prime is a jump prime [Ch14,Theorem 1(2)]. The reader might want to consult the article [EJ14] of the second and third authors to get an impression of how difficult it is to find such examples. It is known due to F. Charles [Ch14,Theorem 1] that these are the only cases in which every good prime is a jump prime.
In this article, we describe a third reason for a prime to jump, the jump character. It was observed experimentally by the first author together with Yu. Tschinkel [CT] that, in the even rank case, one seems to have lim inf BÑ8 γpS, Bq ě 1 2 , for γpS, Bq :" #tp P Π jump pSq | |p| ď Bu #t|p| ď Bu .
We show that this observation is indeed true, except for some corner cases, which are simple to describe. Moreover, the following proves that Π jump pSq contains an entirely regular subset of exact density one half.
Proposition 2.4.2.b) Let S be a K3 surface over a number field K and p Ă O K be a prime of good reduction and residue characteristic ‰ 2. Assume that rk Pic S K is even. Then the following is true.
If det Frob p | T "´1 then rk Pic S p ě rk Pic S K`2 . Here, T Ă H 2 et pS K , É l p1qq denotes the transcendental part of the cohomology.
Furthermore, detpFrob p : T ýq " detpFrob p : H 2 et pS K , É l p1qq ýq detpFrob p : Pic S K ýq and the behaviour of both determinants on the right hand side is completely described by quadratic characters.
The discriminant of the Picard representation. Concerning Pic S K , there is a finite extension field L{K, over which all invertible sheaves are defined. An application of the smooth base change theorem inétale cohomology shows that L is unramified at every prime p, at which S has good reduction, cf. Lemma 2.3.5. Moreover, the field of definition of the maximal exterior power Λ max Pic S K is at most a quadratic subfield Kp a ∆ Pic pSqq Ď L. We call the quantity ∆ Pic pSq, which is determined up to multiplication by squares in K˚, the discriminant of the Picard representation or just the algebraic part of the discriminant of S. One has detpFrob p : Pic S K ýq "ˆ∆ Pic pSq p˙.
The twofoldétale covering describing the determinant of the Frobenius.
The determinant of Frob p on the total H 2 et pS p , É l p1qq may be controlled, as well.
This is in fact an application of a few standard results onétale cohomology, mainly from [SGA4]. Indeed, associated with a smooth and proper family π : F Ñ X of schemes and an even integer i, there is a twofoldétale covering ̺ " ̺ π : Y Ñ X that splits over a closed point z P X exactly when the determinant of Frob on H í et pF z , É l pi{2qq is p`1q.
This was known to experts perhaps already in the days when [SGA4] was written. However, according to our knowledge, no one ever mentioned it in print before T. Saito's article [Sa], published in 2012. In Theorem 4.1.3, we give a formulation more general than T. Saito's, including non-middle cohomology. Furthermore, if the family π allows a compactification over a "reasonable" base scheme P Ą X then ̺ extends to a possibly ramified covering, given by w 2 " ∆, for a section ∆ such that supp div ∆ Ď P zX. For more details, see Lemma 4.3.2.
In particular, if X :" Spec K for K a number field then the above result says the following. Associated with a smooth and proper K-scheme S and an even integer i, there is a field extension L S " Kp a ∆ H i pSqq, at most quadratic, such that detpFrob p : H í et pS K , É l pi{2qq ýq "ˆ∆ H i pSq p˙. I.e., the decomposition law of L S controls the determinants of the Frobenii.
The jump character. For S a K3 surface and i " 2, this yields the following.
Theorem 2.4.4. Let K be a number field and S a K3 surface over K. Moreover, let p Ă O K be a prime of good reduction and residue characteristic ‰ 2. a) Then the following two equations hold. b) If rk Pic S K is even then ∆ H 2 pSq∆ Pic pSq p˙"´1 ùñ rk Pic S p ě rk Pic S K`2 . (2) In other words, t p | p inert in Kp a ∆ H 2 pSq∆ Pic pSqq u Ď Π jump pSq.
The quadratic characterˆ∆ H 2 pSq∆ Pic pSqṁ ight be called the transcendental character of the K3 surface S. Nevertheless, having the argument (2) in mind, we prefer to call it the jump character of S, at least in the even rank case. It may, of course, happen that the jump character is trivial, even for surfaces defined over É, cf. Examples 2.6.11 and 5.3.6. These are the corner cases mentioned above.
By construction, the jump character is unramified at every prime of good reduction. It ramifies, however, at a bad prime, provided that the singular reduction is of the mildest possible type.
Corollary 2.5.4.ii) Let K be a number field and S a K3 surface over K. Moreover, let p Ă O K be a prime of residue characteristic different from 2. If S has a regular, projective model S over O K,p , the geometric fibre S p of which has exactly one singular point being an ordinary double point, then the jump character ∆ H 2 pSq∆ Pic pSq˘r amifies at p.
This result is rather surprising from our point of view. In fact, a direct application of the theory of vanishing cycles [SGA7] shows that`∆ H 2 pSq˘r amifies at p. However, the particular geometry of a K3 surface and the assumption that there is only one singular point on the special fibre impose a certain constraint on the vanishing cycle, implying that`∆ Pic pSq˘m ust be unramified, cf. Theorem 2.5.2.
In addition, we present two algorithms to compute the characters for a given surface S over É, a deterministic one for ∆ H 2 pSq and a statistical one for the jump character. It seems that, for a deterministic computation of the jump character, one either needs at least one bad prime to which Corollary 2.5.4 applies or must have some information on Pic S É available.
Remark 1.1. It turns out that the jumps caused by the jump character do not add up with jumps for reason i) or ii). I.e., there do not need to be bigger jumps in the cases, where two reasons apply. Cf. Remark 2.7.2 and Example 2.7.3. Thus, in the main body of this article, we restrict our considerations to projective K3 surfaces of even geometric Picard rank and assume that real multiplication does not occur.
The criterion for non-triviality of the twofoldétale covering.
For the relative situation, we provide a criterion showing that theétale covering ̺ π is indeed ramified over the boundary P zX, under certain good circumstances. A sufficient condition is that the total scheme F is non-singular and that there is a geometric fibre over a point on the boundary that has exactly one singular point being an ordinary double point (Theorem 4.2.3). The non-triviality of ̺ π again follows from the theory of vanishing cycles and monodromy, more concretely, from the Picard-Lefschetz formula [SGA7,Exp. XV,Théorème 3.4].
As a corollary, we show that the jump character of the fibre F z depends on the base point z P XpKq like p h jump q, for a rational function h jump . Under a fairly general hypothesis, h jump is not just a product of a constant with a perfect square (Corollary 4.4.1). Within such a family, surfaces with trivial jump character are extremely rare. Cf. Remark 5.3.7 at the very end of the article for figures for a concrete family.
The relation to discriminants. As an application of the non-triviality criterion, we present theoretical insight into the nature of the quantity ∆ H i pSq P K˚{pK˚q 2 . In fact, when S is a fibre of a reasonable family then ∆ H i pSq is the reduction modulo pK˚q 2 of a suitably normalised discriminant of S.
Concerning discriminants, there are several not completely equivalent concepts appearing in the literature. We consider the situation of a family π : F Ñ X of proper schemes with non-singular generic fibre. Then the discriminant locus D Ă X is characterised by the property that x P D if and only if F x is singular. If D is pure of codimension one then the discriminant is supposed to be a global section ∆ of an invertible sheaf D P Pic X vanishing exactly at D. The orders of vanishing should to some extent reflect the level of degeneration of the singular fibres. In simple cases, ∆ vanishes everywhere of order 1. Classically, one considers families over base schemes that are proper over a field K. Then the discriminant is determined up to a scaling factor from K˚.
A case, in which the discriminant is known to exist, is that of complete intersections of fixed multidegree and dimension [Ben12,Corollaire 7.3.3] (cf. [GKZ,Ch. 1,Example 4.15] and [Bo1,Bo2]). Moreover, from the theory presented in [Ben12,Chapitre 7], it is evident that one may as well work with -schemes instead of schemes over a base field. This yields that, for the discriminant, one has a canonical choice up to sign.
Thus, let us fix positive integers n and d 1 , . . . , d c ě 2, for 1 ď c ď n, and put V :" Proj Sym À 1ďiďc H 0 pP n , Opd i qq _ . The scheme V parametrises intersections of hypersurfaces of degrees d 1 , . . . , d c in P n . It contains the discriminant locus, an irreducible divisor D Ă V corresponding to the complement of the smooth complete intersections of codimension c. In this situation, we show the following.
Theorem 5.2.3. Let i " n´c be even. Then there is a section ∆ P O V pDq such that div ∆ " pDq and the following holds. Let K be a number field and x P pV zDqpKq be any K-rational point. Then, for any prime p Ă O K of good reduction and residue characteristic ‰ 2, The proof of this result requires an understanding of the relative situation, i.e. of the twofoldétale covering ̺ π : Y Ñ V associated with the family π : F Ñ V , which is in fact given by the equation w 2 " ∆.
An application. The sign of the discriminant in the even-dimensional case. More generally in the case of even dimensional fibres, the covering ̺ π : Y Ñ X associated with the family π provides a canonical choice for the sign of the discriminant. This leads us to the normalised discriminant. Indeed, ̺ π is given by w 2 " ∆, for a section ∆ fulfilling the requirements on a discriminant, as formulated above. This, in itself, is an application of the criterion for non-triviality. Moreover, w 2 "´∆ defines a very different twofold covering. Cf. Definition 5.1.2 for more details.
Remark 1.2. In the particular case of a hypersurface, there is a completely different approach to the discriminant, based on a very general concept of a resultant [Dem,Section 5], cf. [GKZ,Chapter 3]. M. Demazure's divided discriminant [Dem,Définition 4] also fulfils the general requirements for a discriminant.
However, in that approach, the resultant naturally appears as a generator of a principal ideal in a certain -algebra [Dem,Définition 3]. Thus, any selection of a sign for it is arbitrary. The choice that is typically made yields that the diagonal form X d 0`. . .`X d n`1 has a positive discriminant [Dem,Section 5,Exemple 1]. This provides a choice of sign for the discriminant of the hypersurface itself if the dimension n is even or the degree d is odd.
In the even-dimensional case, the divided discriminant is known to differ from the normalised discriminant by a factor of p´1q d´1 2 , when d is odd, and p´1q d 2 n`2 2 , when d is even [Sa,Theorem 3.5].
Remark 1.3. Let us emphasise here that, while the existence of the normalised discriminant relies on S being a member of a reasonable family, the quantity ∆ H i pSq P K˚{pK˚q 2 always exists, as long as S is a smooth and proper K-scheme.
Examples where the jump character is trivial. A Kummer surface associated with the product of two elliptic curves, both being defined over K, has trivial jump character (Example 2.6.9). In order to find examples that have lower geometric Picard rank, we make use of the connection of ∆ H 2 pSq with the classical discriminant ∆pSq, established in Theorem 5.2.3. As it is not at all trivial to compute the discriminant for a general quaternary quartic (in contrast to the case of a ternary quartic), we restricted our considerations to quartics of the special type S : cX 4 3`f 2 X 2 3`f 4 " 0. For these, we show that ∆pSq agrees, up to square factors, with cδpf 4 q, where δ denotes the divided discriminant of a ternary quartic. Cf. Proposition 5.3.1 for a more precise statement.
Another application. Rational curves on even rank K3 surfaces. As another application, we provide a criterion for the existence of an infinite number of rational curves on a K3 surface of even Picard rank. For the odd rank case, this is already known in general, due to the work of J. Li and Ch. Liedtke [LL], which in itself is based on ideas of F. Bogomolov, B. Hassett, and Yu. Tschinkel [BHT].
From the arguments given in [LL], one may in fact deduce the following implication, There are infinitely many primes such that rk Pic S p ą rk Pic S K and S p is non-supersingular ùñ S K contains infinitely many rational curves , cf. [Ben15,remarks after Corollaire 3.10]. If the quantity ∆ H 2 pSq∆ Pic pSq is a non-square in K then our main result provides infinitely many primes such that rk Pic S p ą rk Pic S K . It is a technical issue to what extent it may happen that S p is always supersingular, for all but finitely many of these primes. We show the following.
Theorem 3.1. Let K be number field and S a K3 surface over K. Assume that rk Pic S K is even, that S K has neither real nor complex multiplication, and that ∆ H 2 pSq∆ Pic pSq is a non-square in K. Then S K contains infinitely many rational curves.

Computations.
All the computations are done using magma [BCP], sage [St], and C++ including the libraries FLINT [HJP] and NTL [Sh].

The jump character
Convention. By a character, we always mean a continuous homomorphism from a topological group to a discrete abelian group.
2.1. The determinant of Frob and the relationship to the sign in the functional equation. Let S be a smooth proper variety over a finite field q of characteristic p ą 0. Then the geometric Frobenius Frob operates linearly on the l-adic cohomology modules H í et pS q , l pjqq. The characteristic polynomial Φ piq j of Frob is independent of the choice of l ‰ p and has in fact rational coefficients [DeWI,Théorème (1.6)]. In particular, the determinant of Frob is a rational number and independent of l ‰ p.
In this section, we discuss the behaviour of det Frob. The facts below show that each result on det Frob may be translated into a result about the sign in the functional equation.
Facts 2.1.1 (Deligne, Suh). a) The polynomial Φ piq j P ÉrTs fulfils the functional It is independent of the Tate twist, i.e. of the choice of j. c.i) If i is even then detp´Frob: H í et pS q , É l pi{2qq ýq is either p`1q or p´1q.
In other words, it gives the sign in (3) exactly. ii) If i is odd then N is even and in (3) the plus sign always holds.
Proof. a) and b) The polynomials on both sides of (3) have the same roots as, with z, the number z " q i´2j z is a root of Φ, too, and has the same multiplicity. To show that they perfectly agree, let us adopt the convention that Φ is monic. Then the leading coefficient of the polynomial on the left hand side, which is equal to the constant term of Φ, is nothing but the determinant of p´Frobq on H í et pS q , É l pjqq. As this is a rational number and of absolute value q N 2 pi´2jq , a) follows together with the first assertion of b). The final claim is clear, since c.i) As det Frob " p´1q N Φp0q for Φ monic, this can be read off the functional equation T N Φp1{T q "˘ΦpT q.
ii) If S is projective then, by Poincaré duality and the hard Lefschetz theorem [DeWII,Théorème (4. [EJ15] of the second and third authors. In the proof above, the only property that was used is that every complex root of Φ piq j is of absolute value q i{2´j . This was first proven by P. Deligne in [DeWI,Théorème (1.6)] for the projective case and later in [DeWII,Corollaire (3.3.9)], in general. The assertion had been formulated by A. Weil as a part of his famous conjectures.
2.2. The discriminant of the H i -representation. Let us fix notation and then start by recalling some facts on l-adic cohomology.
Notation 2.2.1. Let K be a field of characteristic ‰ 2 and L{K an at most quadratic field extension. Then, according to Kummer theory, there exists a unique class ∆ L P K˚{pK˚q 2 such that L " Kp a uq for any u P ∆ L . i) In this situation, we shall also write Kp ? ∆ L q for Kp a uq. ii) Assume that K is a number field and p Ă O K is a prime ideal of residue characteristic ‰ 2, at which L{K is unramified. Then the quadratic residue symbol p u p q [Ne, Chapter V, Proposition (3.5)] is independent of the choice of a p-adic unit u P ∆ L . We shall therefore write p ∆ L p q instead of p u p q. Facts 2.2.2. Let K be a field and S a smooth and proper K-scheme. a) Then, for all prime numbers l ‰ char K and all integers i and j, associated with the one-dimensional É l -vector space Λ max H í et pS K , É l pjqq, there is a character of the absolute Galois group of K, which we denote by rdet H i pS K , É l pjqqs : GalpK{Kq ÝÑ Él .
b) Suppose that i is even and that S is pure of dimension i. Then the character rdet H i pS K , É l pi{2qqs is independent of l and has values in t1,´1u Ă Él .
Proof. a) This follows from the fact that every σ P GalpK{Kq induces an automorphism of schemes of S K . b) By Poincaré duality [SGA4, Exp. XVIII, Théorème 3.2.5], there is a canonical non-degenerate symmetric pairing s : H í that is compatible with the operation of GalpK{Kq. According to a standard fact from linear algebra [Wa,Def. 2.9], the pairing s induces another symmetric pairing that is again non-degenerate. The operation of GalpK{Kq is orthogonal with respect to Λ s , which implies that the character rdet H i pS K , É l pi{2qqs must have values in t1,´1u Ă Él .
Let now l ‰ l 1 be two prime numbers. By the Chebotarev density theorem, the conjugacy classes of the Frobenius elements Frob p , for p running through the good primes of residue characteristic ‰ l, l 1 , are dense in GalpK{Kq. It therefore suffices to show rdet H i pS K , É l pi{2qqspFrob p q " rdet H i pS K , É l 1 pi{2qqspFrob p q. For this, recall that one has the canonical isomorphism H í well as its l 1 -adic analogue, according to the smooth specialisation theorem for cohomology groups [SGA4,Exp. XVI,Corollaire 2.3]. The assertion thus follows from [DeWII,Corollaire (3.3.9)].
Definition 2.2.3. i) In the situation of part b), we denote by L S the extension field of K that corresponds to ker rdet H i pS K , É l pi{2qqs under the Galois correspondence.
ii) If char K ‰ 2 then we denote the class in K˚{pK˚q 2 , corresponding to the field extension L S {K, by ∆ H i pSq and call it the discriminant of the H i -representation of S.
Lemma 2.2.4. Let K be a number field and S a smooth and proper K-scheme that is pure of even dimension i. Moreover, let p Ă O K be a prime such that S has a smooth model over O K,p . Then the field L S is unramified at p.
Proof. Assume, to the contrary, that L S {K would ramify at p. Choose a prime l different from the residue characteristic of p and a prime q lying above p. Then the inertia group GalppL S q q {K p q Ă GalpL S {Kq acts faithfully on Λ max H í Take a non-trivial element σ P GalpL q {K n p q Ď GalpL q {K p q, for K n p :" L q X K nr p the maximal unramified subfield of L q . Then σ acts non-trivially on Λ max H í et pS K , É l pi{2qq. This, however, contradicts the smooth specialisation theorem. Indeed, GalpK p {K p q must operate on H í et pS K , É l pi{2qq, and therefore all the more on Λ max H í et pS K , É l pi{2qq, via its quotient GalpK nr Remarks 2.2.5. i) In particular, for every prime ideal p of good reduction and residue characteristic ‰ 2, one has ii) For any number field K 1 Ě K, one has ∆ H i pS K 1 q " ∆ H i pSq¨pK 1˚q2 P K 1 {pK 1˚q2 .
iii) The name "discriminant" has not been chosen at random. Indeed, Facts 2.2.2 allow a generalisation to families over a general base scheme. The quadratic field extension then goes over into a twofoldétale covering ̺ π : Y Ñ X, ramified at most over the discriminant locus of the family. If π is sufficiently reasonable then ̺ π is given by w 2 " ∆, for ∆ a normalised version of the discriminant of the family π. For a member S " F x of the family, ∆pxq belongs to the class ∆ H i pSq. Cf. Section 5 for more details.
Facts 2.3.1. Let K be a field and S a smooth projective surface over K. Then, associated with the one-dimensional É-vector space Λ max pNSpS K q b Éq, there is a character of the absolute Galois group of K, which we denote by rdetpNSpS K q b Éqs: GalpK{Kq ÝÑ t1,´1u Ă É˚.
In particular, it defines an at most quadratic field extension L alg S {K. Proof. Every σ P GalpK{Kq induces an automorphism of the Néron-Severi group NSpS K q. This defines the character rdetpNSpS K q b Éqs, at first with values in É˚.
Moreover, by the Hodge index theorem, there is a canonical non-degenerate symmetric pairing s : pNSpS K q b ÉqˆpNSpS K q b Éq Ñ É that is compatible with the operation of GalpK{Kq. According to [Wa,Def. 2.9], the induced pairing Λ s : Λ max pNSpS K q b ÉqˆΛ max pNSpS K q b Éq ÝÑ É is again symmetric and non-degenerate. The operation of GalpK{Kq is orthogonal with respect to Λ s , which implies that the character rdetpNSpS K q b Éqs must have values in t1,´1u Ă É˚. b) For surfaces such that H 1 pS K , É l q " 0, one has PicpS K q b É -NSpS K q b É.
In this case, one may write ∆ Pic pSq instead of ∆ NS pSq and speak of the discriminant of the Picard-representation.
Remark 2.3.3. The algebraic part of the discriminant of S should not be confused with the discriminant of NSpSq or NSpS K q as a lattice. One might think about it instead as follows.
Since NSpS K q b É is a finite-dimensional É-vector space, there is a finite field extension of K, over which all elements of NSpS K q b É are defined. We call the smallest such field L the field of definition of NSpS K qb É. Then GalpK{Kq operates on NSpS K q b É via its quotient GalpL{Kq and the operation of that is faithful.
Therefore, the operation of GalpK{Kq on Λ max pNSpS K q b Éq factors via GalpL{Kq, too. On the other hand, Λ max pNSpS K q b Éq is one-dimensional, so that the operation must factor via t1,´1u. The stabiliser is of index at most 2 and the field of definition of Λ max pNSpS K q b Éq is an at most quadratic extension Kp a ∆ NS pSqq " L alg S Ď L. Remark 2.3.4. If K 1 Ě K is a number field extending K then one has ∆ NS pS K 1 q " ∆ NS pSq¨pK 1˚q2 P K 1 {pK 1˚q2 . Lemma 2.3.5. Let K be a number field and S a smooth projective surface over K. Moreover, let p Ă O K be a prime such that S has a smooth model over O K,p .
Then the field of definition of NSpS K q b É is unramified at p. Proof. Write L for the field of definition of NSpS K q b É and assume that L would ramify at p. Choose a prime q lying above p. Then the inertia group GalpL q {K p q Ă GalpL{Kq acts faithfully on NSpS K q b É.
Choose a non-trivial element σ P GalpL q {K n p q Ď GalpL q {K p q, for K n p the maximal unramified subfield of L q . Then, for any prime l, σ operates non-trivially on the image of the first Chern class homomorphism This, however, is in contradiction with the smooth specialisation theorem for cohomology groups, as seen before. The fact that the Chern class homomorphism factors via the Néron-Severi group, i.e. via numerical equivalence, follows from Matsusaka's theorem [Ma,Theorem 4], cf. [An,paragraph 3.2.7].
Remarks 2.3.6. i) In particular, Kp a ∆ NS pSqq " L alg S is unramified at every prime ideal p of good reduction. If, moreover, the residue characteristic is ‰ 2 then one has ii) Somewhat surprisingly, in the particular case of a K3 surface, the assertion of Lemma 2.3.5 is still true when S has bad reduction at p of the mildest possible form.

K3 surfaces.
There is a strong relation, which relies on Tate's and Serre's conjectures in general, but is established for K3 surfaces, between the Galois operation on l-adic cohomology and the variation of the geometric Picard ranks under reduction modulo various primes [EJ12b,CT]. From our point of view, this is in fact the main application of the constructions presented so far.
Facts 2.4.1. Let S be a K3 surface over a base field K. a) Then Pic S K is a free abelian group of rank at most 22. If K is of characteristic zero then the rank is at most 20. If K is finite of characteristic ‰ 2 then the rank is even. b) If K is finite of characteristic ‰ 2 then rk Pic S K is equal to the number [counted with multiplicities] of all eigenvalues of Frob on H 2 et pS K , É l p1qq that are roots of unity.
Proof. a) If c P Pic S K is such that xc, cy " 0 then, according to Riemann-Roch, h 0 pcq`h 0 p´cq ě 2. If, in particular, c ‰ 0 then xc, hy ‰ 0, for h a hyperplane section. This argument immediately shows that Pic S K is torsion-free. The assertions on the rank immediately follow from this, together with the standard calculations of the cohomology of K3 surfaces (cf. [Bea, Chapter VIII] and [GH,p. 590]). The final claim is a direct consequence of b). b) Every invertible sheaf is defined over a finite extension of K. Hence, for some e ě 1, the power Frob e operates identically on H alg :" c 1 pPicpS K qb É l q. This shows that there are at least rk Pic S K eigenvalues being roots of unity.
On the other hand, let r be the number of eigenvalues that are roots of unity. Choose e ě 1 such that Frob e has the eigenvalue 1 of algebraic multiplicity r. Then the geometric multiplicity is equal to r, too, by [DeK3, Corollaire 1.10]. Moreover, the Tate conjecture has recently been proven for K3 surfaces over finite fields of odd characteristic ([Ch13, Corollary 2], [MP, Theorem 1], cf. [LMS]). Thus, the Picard rank over the extension field K 1 of K of degree e is equal to r.
Furthermore, for every prime number l, we have a canonical orthogonal decomposition Here, H alg " c 1 pPicpS K q b É l q is clearly GalpK{Kq-invariant. Moreover, T :" H K alg is GalpK{Kq-invariant, too, as the Galois operation is orthogonal. Let now p Ă O K be any prime of good reduction and of residue characteristic different from l. Then Frob p P GalpK{Kq is determined only up to conjugation. But this suffices to have well-defined eigenvalues and a well-defined determinant of Frob p , associated with any vector space being acted upon by GalpK{Kq. In particular, Our main theoretical observation on the distribution of the Picard ranks of the reductions is then as follows.
Proposition 2.4.2. Let S be a K3 surface over a number field K and p Ă O K be a prime of good reduction and residue characteristic ‰ 2. a) Then rk Pic S p ě rk Pic S K . b) (Rank jumps) Assume that rk Pic S K is even. Then the following is true. If det Frob p | T "´1 then rk Pic S p ě rk Pic S K`2 . Proof. Choose a prime number l, different from the residue characteristic of p. Then, according to smooth base change, H 2 In particular, by transport of structure, the orthogonal decomposition (6) carries over into Moreover, under the first isomorphism, the operation of GalpK p {K p q is compatible with that of inertia I Ă GalpK{Kq, while the second isomorphism shows that GalpK p {K p q acts via its quotient GalpK nr p {K p q. In particular, the operation of For instance, Frob and Frob p have the same eigenvalues on H alg , as well as on T . a) By Lemma 2.3.5, the field of definition of Pic S K is a number field unramified at p. Therefore, only a finite quotient of GalpK nr p {K p q operates on H alg . In particular, there exists an integer e ą 0 such that Frob e acts identically. Thus, all eigenvalues of Frob on H alg are roots of unity. Fact 2.4.1.c) implies the claim. b) In view of what was just shown, we need to show that Frob operates on T with at least two eigenvalues being roots of unity. For this, we observe that each eigenvalue is of absolute value 1, so that those different from 1 and p´1q come in pairs tz, zu of complex conjugates. As zz " 1 and det Frob p | T "´1, one of the eigenvalues must be equal to p´1q. Finally, as dim T " 22´rk Pic S K is even, a further eigenvalue 1 is enforced. This completes the proof.
Remark 2.4.3. The proof above shows that, in addition to the specialisations of the invertible sheaves from Pic S K , the Picard group of Pic S p has (at least) two further generators. One of them may be chosen to be defined over p , the other over its quadratic extension.
Theorem 2.4.4. Let K be a number field and S a K3 surface over K. Moreover, let p Ă O K be a prime of good reduction and residue characteristic ‰ 2. a) Then the following two equations hold.
Proof. a) The first formula is a particular case of formula (4). The second one is a consequence of the first together with formulae (5) and (7). b) This follows from a), together with Proposition 2.4.2.b).
Corollary 2.4.5. Let K be a number field and S a K3 surface over K. Assume that ∆ H 2 pSq∆ Pic pSq is a non-square in K. Then lim inf BÑ8 γpS, Bq ě 1 2 Definition 2.4.6. For K a number field and S a K3 surface over K, we call ∆ H 2 pSq∆ Pic pSqṫ he jump character of S.

2.5.
The criterion for non-triviality for a single K3 surface.
Proposition 2.5.1 (The vanishing cycle). Let K be a number field and S a K3 surface over K. Moreover, let p Ă O K be a prime such that S has a regular, projective model S over O K,p , the geometric fibre S p of which has exactly one singular point z being an ordinary double point. Then, for every prime number l, different from the residue characteristic of p, the vanishing cycle [SGA7, Exp. XV, Théorème 3.4.(i)] associated with z, fulfils First of all, one has xδ z , δ z y "´2 [SGA7, Exp. XV, Théorème 3.4.(i)]. Moreover, the monodromy operator V : H 2 et pS K , É l p1qq ý, the operation of a particular non-trivial element of the inertia group I p Ă GalpK{Kq on the cohomology of S K , is described by the Picard-Lefschetz formula [SGA7,Exp. XV,Théorème 3.4
In particular, V pδ z q "´δ z . Now assume, to the contrary, that δ z R H K alg . I.e., that xc, δ z y ‰ 0 for a certain class c P H alg . Then formula (9) immediately implies that δ z " 1 xc, δ z y rV pcq´cs P H alg .
I.e., there is some d P PicpS K q such that c 1 pdq " δ z . As xd, dy "´2, Riemann-Roch shows that h 0 pdq`h 0 p´dq ě 1. In other words, either d or p´dq is represented by an effective divisor. But then V cannot interchange the two, a contradiction.
Theorem 2.5.2 (K3 surfaces with reduction to one ordinary double point). Let K be a number field and S a K3 surface over K. Moreover, let p Ă O K be a prime such that S has a regular, projective model S over O K,p , the geometric fibre S p of which has exactly one singular point being an ordinary double point.

i) Then the field of definition of PicpS
ii) If the residue characteristic of p is different from 2 then the character`∆ Pic pSqȋ s unramified at p, while`∆ H 2 pSq˘r amifies.
Proof. i) Choose a prime number l, different from the residue characteristic of p.
Then there is the short exact sequence [SGA7, Exp. XV, Théorème 3.4.(ii)] provided by the theory of vanishing cycles. Together with the result of Proposition 2.5.1, it shows that every invertible sheaf on S K extends to S p . Hence, the field of definition of PicpS K q b É is contained in K nr p . ii) The first claim is a direct consequence of i). For the second, observe that V fixes all cohomology classes perpendicular to δ z and sends δ z to p´δ z q. Therefore, detpV : H 2 et pS K , É l p1qq ýq "´1. In particular, the field corresponding under the Galois correspondence to ker rdet H 2 pS K , É l p1qqs is not contained in K nr p . Remarks 2.5.3. i) The regularity of the model S implies that the singular point on S p does not lift to a O nr Kp -rational point on S. ii) When there are two singular points instead of one then the field of definition of PicpS K q b É may well ramify. Plenty of examples may be deduced from Lemma 5.3.5.a), below. The argument above then just shows that a non-trivial linear combination of δ z 1 and δ z 2 lies in H K alg . Corollary 2.5.4 (The jump character). Let K be a number field and S a K3 surface over K. Moreover, let p Ă O K be a prime of residue characteristic different from 2. i) If S has good reduction at p then`∆ H 2 pSq∆ Pic pSq˘i s unramified at p.
ii) If S has a regular, projective model S over O K,p , the geometric fibre S p of which has exactly one singular point being an ordinary double point, then the jump character`∆ H 2 pSq∆ Pic pSq˘r amifies at p. Proof. i) is clear from Lemmata 2.2.4 and 2.3.5, while the assertion of part ii) follows from Theorem 2.5.2.ii).
2.6. Examples and experimental results.
Algorithm 2.6.1 (Computing ∆ H 2 pSq). Given the set tq 1 , . . . , q m u of all bad primes of S and an oracle for detpFrob p : H 2 et pS É , É l p1qq ýq for any p ‰ q j , this algorithm computes ∆ H 2 pSq, for S a proper surface over É. Remarks 2.6.2. i) The oracle for detpFrob p : H 2 et pS É , É l p1qq ýq is, of course, provided by counting the points on S that are defined over p and some of its extensions. ii) Dirichlet's Theorem on primes in arithmetic progressions ensures that there exist primes so that the matrix A has rank m`1. iii) Assume that the surface S is K3. For some or many of its bad primes, Theorem 2.5.2 applies. The solution vector is then bound to component 1 at the corresponding coordinates. If S has a model in some P N that is given by explicit equations then one may determine the bad primes using Gröbner bases and integer factorisation, and then analyse the singular points. Our experience is that this often leads to an enormous gain for a "random" surface, while for the constructed examples, we present below, it would not help much. iv) There is an obvious modification of Algorithm 2.6.1 to directly determine the jump character.
Algorithm 2.6.3 (Statistical algorithm computing the jump character). Given the set tq 1 , . . . , q m u of all bad primes of S and a list of non-jump primes, this algorithm gives information on ∆ H 2 pSq∆ Pic pSq, for S a K3 surface over É.
i) Add q 0 :"´1 to the list of bad primes. ii) Build a matrix A, the entries of which are the Legendre symbols p q j p i q, for the non-jump primes p i . iii) Determine the kernel of A. Calculate a candidate for the jump character from each kernel vector.
Remarks 2.6.4. i) If the kernel is the null space then this proves the jump character to be trivial. If the kernel is one-dimensional then there are two possible answers. A non-trivial character, which is directly computed from a kernel vector, and the trivial one.
ii) If the kernel is still one-dimensional when the system of equations is rather overdetermined then this gives strong evidence for the jump character to be non-trivial. In practice, we work with at least 4pm`1q random primes.
iii) As soon as it applies, Corollary 2.5.4.ii) excludes the trivial character, and therefore makes the outcome of Algorithm 2.6.3 usually unique. It is useful as well to accelerate the calculations.
Example 2.6.5. Let S be the diagonal quartic in P 3 É , given by X 4 0`X 4 1`X 4 2`X 4 3 " 0. Then the geometric Picard rank of S is 20 and the jump character is given by p´1 q.
On the other hand, it is classically known that the 48 lines on S É generate the geometric Picard group, which is of rank 20. In particular, Pic S É is defined over 2q. Moreover, [Br,Appendix A,Examples A62,B33,C27,and D27] show that the Galois representation PicpS É q b splits into characters as Here, χ K denotes the non-trivial character that becomes trivial after restriction to GalpÉpζ 8 q{Kq. Consequently, Λ max PicpSq b " χ Épiq and ∆ Pic pS É q "´1.
Remark 2.6.6. It is known at least since 1963 [T] that, in this example, there are no rank jumps, except for those explained by the jump character. I.e., one has rk Pic S p " 20 for all primes p " 1 pmod 4q. In fact, the eigenvalues of Frob p on H 2 et pS É , É l p1qq may be determined using Jacobi sums [IR,Chapter 8,Theorem 5] and it turns out that two of them are π 2 p and its conjugate, for p " ππ a factorisation in Épiq. Cf. [PS,particularly formulae (12) and (13)] for more details.
Example 2.6.7. Let S be the double cover of P 2 É , given by w 2 " X 6 0`X 6 1`X 6 2 . Then the geometric Picard rank of S is 20 and the jump character is given by p´3 q.
The ramification sextic allows 18 tritangent lines of the type X i`ζ m 12 X j " 0, for m odd. The components of their preimages do not yet generate the geometric Picard group, not even up to finite index. They do, however, together with the preimages of the conics of the type X i X j`ζ m 6 3 ? 2 X 2 k " 0, which are six times tangent to the ramification sextic.
We implemented in magma a function to compute intersection numbers on S and, starting with 14 tritangent lines and six conics being six times tangent, found a non-degenerate 20ˆ20 intersection matrix. Using this, it turns out that the field of definition of Pic S É is in fact Épζ 3 , 3 ?
Remark 2.6.8. Again, there are no rank jumps, except for those explained by the jump character. I.e., one has rk Pic S p " 20 for all primes p " 1 pmod 3q.
The eigenvalues of Frob p may again be determined using Jacobi sums. Here, it turns out that two of them are Jpω, ω, ωq{p [IR,Proposition 8.5.1] and its conjugate, for ω a primitive sextic character on p . A short calculation, using [IR,Chapter 8,Theorem 3] and [BE, Theorems 3.1 and 3.4)] shows that these two eigenvalues evaluate to p´1 p q π 2 p and its conjugate, for π the primary element in Épζ 3 q of norm p.
Example 2.6.9. Let K be number field and S be the Kummer surface of an abelian surface over K that geometrically splits into a product E 1ˆE2 of elliptic curves. Assume that rk Pic S K " 18. a) If the elliptic curves E 1 and E 2 are defined over K then the jump character of S is trivial. b) If the elliptic curves E 1 and E 2 are defined over a quadratic extension Kp ? dq and conjugate to each other then the jump character of S is p and both factors are trivially acted upon by GalpK{Kq. b) Let σ P GalpK{Kq be any automorphism that changes the sign of ? d. Then σ interchanges the components of H 1 I.e., σ operates with eigenvalues p´1q and 1, both of multiplicity 2. Hence, on on the algebraic part. Therefore, the eigenvalues on T are p´1q with multiplicity 3 and 1 with multiplicity 1. Hence, every σ P GalpK{Kq as chosen above operates as p´1q on Λ max T , which is enough to imply the assertion. Each of the factors listed is reported as being prime by magma, version 2.21.8.
Example 2.6.11. Let S be the K3 surface over É, given by the equation for f 2 pX 0 , X 1 , X 2 q :" X 2 0´X 0 X 1´X0 X 2´X1 X 2 and f 4 pX 0 , X 1 , X 2 q :"´X 3 0 X 2`X0 X 2 1 X 2´X 4 1´X 4 2 . Then the geometric Picard rank of S is 8 and the jump character of S is trivial.
Proof. We show in Lemma 5.3.4 that space quartics that are of the form (10) are of geometric Picard rank at least 8. Thus, for the first claim, it suffices to find a prime p of good reduction such that rk Pic S p " 8. For example,p " 19,43,61,101,109,139,149,151,157, and 163 do the job, as is easily shown in the usual way, based on counting points. Cf. [CT] for more details and further references.
On the other hand, a calculation using Gröbner bases shows that S has bad reduction only at the primes 2, 3, 47, and 431. Using Algorithm 2.6.3, one then proves the triviality of the jump character. In fact, only the first five non-jump primes 19, 43, 61, 101 and 109 are needed in order to do this.
Example 2.6.12. Let S be the K3 surface over É, given by the equation for f 4 pX 0 , X 1 , X 2 q :" X 4 0´X 3 0 X 1´2 X 3 0 X 2´X 2 0 X 1 X 2`X0 X 2 1 X 2´X 4 1´X 4 2 . Then the geometric Picard rank of S is 8 and the jump character of S is p´1 . q or trivial.
Moreover, a calculation using Gröbner bases shows that S has bad reduction only at the primes 2, 7, 6449, and 39 870 353. Algorithm 2.6.3 then easily proves the claim on the jump character. The first four non-jump primes 5, 13, 41, and 53 are in fact sufficient.
Remark 2.6.13. We will come back to Examples 2.6.11 and 2.6.12 in 5.3.6, where we prove that the jump character is indeed non-trivial in the second one and give a by far more systematic proof for triviality in the first. For this, we have to develop the theory further, which means to consider the variation of a variety in a complete family π : F Ñ X and to make precise the relation of the ∆ H i pF x q to the discriminant of the family. And, finally, to compute the discriminant, at least for quartics that are special of the form (10).

Interaction of jumps.
Lemma 2.7.1 (Degree two K3 surfaces). Let K be number field and S a K3 surface over K, given by an equation of type w 2 " f 6 pX 0 , X 1 , X 2 q, for f 6 a homogeneous form of degree 6. Write S λ : λw 2 " f 6 pX 0 , X 1 , X 2 q for the quadratic twist by λ P K˚. Then ∆ H 2 pS λ q " λ∆ H 2 pSq and ∆ Pic pS λ q " λ rk Pic S K´1 ∆ Pic pSq .
Proof. Let p be a good prime of S such that λ is a p-adic unit. Then, for the reductions mod p, one has that pS λ q p is a non-trivial twist of S p in the case that λ is a non-square modulo p, and S p -pS λ q p , otherwise. The assertion therefore follows from [EJ10,Fact 25].
Remark 2.7.2 (The odd rank case). Assume that rk Pic S K " 1. Then, for any prime p of good reduction, there exists some p-adic unit λ P K˚such that ∆ H 2 pS λ q∆ Pic pS λ q is a non-square modulo p. If the effect of the odd rank would add up with that of the transcendental character then this would imply rk Pic S p " rk PicpS λ q p ě 4. There are, however, explicit degree 2 K3 surfaces known of geometric Picard rank 1 that reduce to geometric Picard rank 2 at certain primes [vL,  Example 2.7.3 (The case of real multiplication). Let S be the minimal desingularisation of the double cover of P 2 É , given by w 2 " X 0 X 1 X 2¨f3 pX 0 , X 1 , X 2 q, for There is strong evidence that S has real multiplication by Ép ? 3q. Indeed, S is the surface V p3q 1,1 from [EJ16,Conjectures 5.2]. It has bad reduction only at 2, 3, and 5. Modulo all other primes p ă 1000, the reduction S p is of geometric Picard rank 18, except for p " 263, where the geometric Picard rank is 22. On the other hand, a sublattice of Pic S É of rank 16 may be explicitly given. Altogether, taking the real multiplication for granted, one concludes that rk Pic S É " 16.
Concerning Pic S É , there are 13 obvious generators, given by the pull-back of a general line on P 2 É and the exceptional curves obtained by blowing up the twelve singular points of the ramification locus. Ten of these singular points are defined over É, the other two over Ép ?´2 q. Hence, this part of PicpS É q b splits into irreducible components as χ 12 triv ' χ Ép ?´2 q . Further generators are formed by a line and two conics in P 2 É , the preimages of which split in S. From these, one finds that q and, consequently, ∆ Pic pSq " 1. The field of definition of PicpS É q is Épi, 3q. On the other hand, Algorithm 2.6.1 yields ∆ H 2 pSq " 3, such that the jump character is given by p 3 q.
If the effect of real multiplication would add up with that of the transcendental character then this would imply rk Pic S p ą 18 for every prime p such that p 3 p q "´1, a contradiction.

Infinitely many rational curves
As an application of Theorem 2.4.4, we have the following result.
Theorem 3.1. Let K be number field and S a K3 surface over K. Assume that rk Pic S K is even, that S K has neither real nor complex multiplication, and that ∆ H 2 pSq∆ Pic pSq is a non-square in K. Then S K contains infinitely many rational curves.
Remarks 3.2. i) It is conjectured that every K3 surface S over an algebraically closed field K contains infinitely many rational curves. There is a lot of evidence for this conjecture. Proven cases include, most notably, those that the Picard rank is odd or that K is of characteristic zero and X cannot be defined over É. The idea of the proof for the odd rank case is due to F. Bogomolov, B. Hassett, and Yu. Tschinkel [BHT] and was later refined by J. Li and Ch. Liedtke [LL]. An overview on their approach is given in [Ben15]. There are further sufficient conditions, including those that S has infinitely many automorphisms or that S is elliptic. As a consequence, it is known that the conjecture is true for all K3 surfaces S of Picard rank ‰ 2, 4.
ii) The transcendental part of T Ă H 2 pX , Éq, considered as a pure weight-2 Hodge structure, has an endomorphism algebra End Hs pT q that may only be a totally real field or a CM field [Za, Theorem 1.6.a) and Theorem 1.5.1]. Our assumption concerning real and complex multiplication just means that End Hs pT q " É. Lemma 3.3. Let K be a number field and S be a K3 surface over K. Assume that S K has neither real nor complex multiplication. Then, for every quadratic field extension L{K, there are infinitely many inert primes p Ă O K such that the reduction S p is non-supersingular.
Proof. In the case that rk Pic S K " 20, the linear algebra degenerates, so that S K automatically has complex multiplication [SI,Theorem 4]. We may therefore assume that r :" dim T ě 3.
We choose a prime l and put T l Ă H 2 et pS É , É l q to be the transcendental part of l-adic cohomology. As End Hs pT q " É, we know that the image of the canonical continuous representation ̺ l : GalpK{Kq Ñ GOpT l , x. , .yq is Zariski dense. Indeed, this follows from the Mumford-Tate conjecture, proven by S. G. Tankeev [Ta90,Ta95], together with Yu. G. Zarhin's explicit description of the Mumford-Tate group in the case of a K3 surface [Za, Theorem 2.2.1]. Now, let us assume, to the contrary, that for all but finitely many inert primes p, the reduction S p would be supersingular. We put M :" tp Ă O K prime ideal | p is a prime field, # p ‰ l, p inert in L, p good for S, S p supersingularu . Then M Ď I, for I the set of all inert primes, and IzM is of analytic density zero. For every prime p Ă O K , we choose a Frobenius element Frob p P GalpK{Kq. By the Chebotarev density theorem, the elements σ´1 Frob p σ P GalpK{Kq, for p P M and σ P GalpK{Kq, are topologically dense in the non-trivial coset of GalpK{Kq modulo GalpK{Lq. Thus, there are two elements σ 1 , σ 2 P GalpK{Kq such that t σ j σ´1 Frob p σ | j " 1, 2, p P M, σ P GalpK{Kq u is dense in GalpK{Kq.
On the other hand, for p P M one has p | Tr Frob p,T l and det Frob p,T l "˘p r , when putting p :" # p . As | Tr Frob p,T l | ď rp, this shows that pTr Frob p,T l q r "˘k r det Frob p,T l for´22 ă´r ď k ď r ă 22. Condition (12), in itself, defines a Zariski closed subset C Ă GOpT l , x. , .yq that is invariant under conjugation. As GOpT l , x. , .yq É l -GO r,É l for r ě 3, one easily sees that dim C ă dim GOpT l , x. , .yq. Thus, the union σ 1 C Y σ 2 C cannot be the whole group. Consequently, the image of GalpK{Kq Ñ GOpT l , x. , .yq is not Zariski dense, a contradiction.
Proposition 3.4 (cf. [LL,Proposition 4.2]). Let K be a number field and S Ă P N K be a K3 surface. Assume that Pic S " Pic S K and that there is an infinite set J of primes such that rk Pic S p ą rk Pic S for p P J. Then there exist a sequence without repetitions pp j q jPAE of primes from J and a sequence pD p j q jPAE of rational curves D p j Ă S p j such that the following two conditions are satisfied.
The class pD p j q P PicpS p j q does not lie in the image of Pic S K under specialisation, for any j, and lim jÑ8 deg D p j " 8.
Proof. For pp j q jPAE , one may take any sequence without repetitions of good primes from J. Moreover, for each j P AE, we choose an effective divisor D 1 p j Ă S p j not lying in the image of Pic S under specialisation.
By the Theorem of Bogomolov-Mumford "`ε" [LL, Theorem 1.1], there is a sum of rational curves that is linearly equivalent to D 1 p j . One of the summands is not contained in the image of the specialisation. We take that for D p j .
It remains to show that lim jÑ8 deg D p j " 8. Assuming the contrary, we conclude that there must exist a subsequence pp j k q kPAE so that deg D p j k " c is constant. To obtain a contradiction from this, let us consider the Zariski closure S Ă P N There is a Hilbert scheme Mor c pP 1 , S q Ă HilbpP 1 O KˆS pec O K S q being of finite type over O K that parametrises morphisms f : P 1 Ñ S of degree c. As we are interested only in closed points, let us put M :" Mor c pP 1 , S q red . According to the closedness of the singular locus [EGA IV, Corollaire (6.12.6)], there is a filtration by closed subschemes M " M 0 Ą M 1 Ą . . . Ą M k " H such that each difference M e zM e`1 is a non-singular scheme. By [EGA IV, Corollaire (6.8.7)], M e zM e`1 is even smooth over O K , outside finitely many fibres. This implies [EGA IV, Proposition (17.14.1)] that, for j " 0, the rational curves D p j must be specialisations of rational curves on S K , in contradiction to the construction of the D p j .
Proposition 3.5 (Li-Liedtke). Let K be a number field and S Ă P N K be a K3 surface. Assume that there exist a sequence without repetitions pp j q jPAE of primes and a sequence pD p j q jPAE of rational curves D p j Ă S p j satisfying the following conditions. Each S p j is non-supersingular, pD p j q does not lie in the image of the Picard group Pic S K of the generic fibre under specialisation, for any j, and lim jÑ8 deg D p j " 8.
Then, for every j " 0, there exists a rational curve D j Ă S K such that its specialisation to S p j is reducible, containing D p j as one of its components. In particular, Proof. This is shown in the proof of [LL,Theorem 4.3], which is in fact the main achievement of the article [LL] of J. Li and Ch. Liedtke. The argument involves deformation theory and considerations about a whole family of K3 surfaces containing S.
3.6. Proof of Theorem 3.1. As ∆ H 2 pSq∆ Pic pSq is a non-square in K, the field L :" Kp a ∆ H 2 pSq∆ Pic pSqq is indeed a quadratic extension. By Lemma 3.3, we have an infinite set J of inert primes such that S p is non-supersingular for every p P N. Moreover, rk Pic S p ą rk Pic S K according to Theorem 2.4.4.b). Let now K 1 Ě K be the field of definition of Pic S K . For each p P J, there is at least one prime p 1 Ă O K 1 lying above p. This yields an infinite set J 1 of primes in O K 1 , to which Proposition 3.4 applies. It provides a sequence pp j q jPAE of primes in J 1 without repetitions and rational curves D p j Ă S p j , not lying in the image of Pic S K under specialisation, such that lim jÑ8 deg D p j " 8. Knowing this, Proposition 3.5 yields a sequence pD j q jPAE of rational curves on S K of degrees tending towards infinity. This completes the proof.
4. The twofoldétale covering describing the determinant of Frob 4.1. The twofoldétale covering. The goal of this section is to investigate the behaviour of det Frob in families. Our approach is based on the following very general construction.
Facts 4.1.1. Let X be an irreducible scheme and l a prime number that is different from all residue characteristics of X. Choose a geometric point η : Spec KpXq Ñ X lying over the generic point η P X. Finally, let π : F Ñ X be a smooth and proper family of schemes. a) Then, for all integers i and j, i) the higher direct image sheaf R i π˚É l pjq is twisted-constant. ii) Associated with the twisted-constant sheaf Λ max R i π˚É l pjq of rank one, there is a character of theétale fundamental group, which we denote by rdet H i pF, É l pjqqs : πé t 1 pX, ηq ÝÑ Él . b) Let i be an even integer. Unless F η is pure of dimension i, suppose that the family π is projective. Then the character rdet H i pF r 1 l s, É l pi{2qqs has values in t1,´1u Ă Él . In particular, it defines a twofoldétale covering ̺ π : Y Ñ X.
Note that we use the term twisted-constant for what is called "constant tordu constructible" in the terminology of [SGA5,Exp.  twisted-constant É l -sheaves on X of rank one and É l -vector spaces of dimension one together with a continuous πé t 1 pX, ηq-operation. b) There is a canonical non-degenerate symmetric pairing s : R i π˚É l pi{2q| Xr 1 l sˆR i π˚É l pi{2q| Xr 1 l s ÝÑ pÉ l q Xr 1 l s . Indeed, in the case that F η is pure of dimension i, this follows from Poincaré duality [SGA4, Exp. XVIII, Théorème 3.2.5], while otherwise the same is true due to the hard Lefschetz theorem [DeWII,Théorème (4.1.1)]. According to a standard fact from linear algebra [Wa,Def. 2.9], the pairing s induces another symmetric pairing Λ s : Λ max R i π˚É l pi{2q| Xr 1 l sˆΛ max R i π˚É l pi{2q| Xr 1 l s ÝÑ pÉ l q Xr 1 l s that is again non-degenerate in every fibre.
Thus, under the equivalence of categories [SGA5,Exp. 6, Proposition 1.2.5], the sheaf Λ max R i π˚É l pi{2q| Xr 1 l s corresponds to a one-dimensional É l -vector space V together with a non-degenerate symmetric pairing Λ s : VˆV Ñ É l , and a continuous πé t 1 pX, ηq-operation that is orthogonal with respect to Λ s . This implies that the character rdet H i pF r 1 l s, É l pi{2qqs must have values in t1,´1u Ă Él .
Proposition 4.1.2. Let K be a number field, O K its ring of integers, X a normal, integral O K -scheme of finite type, and π : F Ñ X be a smooth and proper family of schemes. Furthermore, let an even integer i be given. Unless F η is pure of dimension i, suppose that π is projective. Fix, finally, a geometric point η : Spec KpXq Ñ X lying over the generic point η.
Then there exists a unique character rdet H i pF qs : πé t 1 pX, ηq Ñ t1,´1u such that, for every prime number l that is invertible in KpXq, the composition with the natural homomorphism πé t 1 pXr 1 l s, ηq Ñ πé t 1 pX, ηq exactly yields rdet H i pF r 1 l s, É l pi{2qqs.
Proof. In view of Fact 4.1.1.b), all one has to show is that the characters rdet H i pF r 1 l s, É l pi{2qqs, for the various values of l, are compatible with each other and define a unique homomorphism from πé t 1 pX, ηq. In the case that F η is pure of dimension i, this is [Sa,Lemma 3.2

] and T. Saito's proof works in general.
Theorem 4.1.3. Let K be a number field, O K its ring of integers, X a normal, integral O K -scheme of finite type, and π : F Ñ X a smooth and proper family of schemes. Furthermore, let an even integer i be given. Unless F η is pure of dimension i, suppose π to be projective. b) The covering ̺ π is given by the character rdet H i pF qs described in Proposition 4.1.2. c) By the property asserted in a), the covering ̺ is uniquely determined up to isomorphism.
Proof. a) and b) We take property b) as a definition for ̺ π . In order to show that ̺ π has the property desired for a), let x P X be any closed point. Choose a geometric point x lying over x and a prime l that is different from the residue characteristic of x. Then, by construction, c) Uniqueness is a direct consequence of the Chebotarev-Lang density theorem [La] (cf. [Ra,Lemma 1.7]), stating that the Frobenius elements Frob x for all closed points x P X generate the abelian quotient of theétale fundamental group πé t 1 pX, ηq. Remark 4.1.4. Note that there exists an isomorphism πé t 1 pXr 1 l s, xq -ÝÑ πé t 1 pXr 1 l s, ηq, simply because Xr 1 l s is connected [SGA1,Exp. V,§7,p. 141]. This isomorphism is unique up to conjugation in πé t 1 pX, ηq so that, strictly speaking, Frob x P πé t 1 pX, ηq is defined only up to conjugation. This is, however, completely sufficient for the purposes of the proof above, as it works entirely with homomorphisms to the abelian group t1,´1u.
Corollary 4.1.5. The formation of ̺ π commutes with arbitrary base change.
Proof. The formation of R i π˚É l pi{2q, and hence that of Λ max R i π˚É l pi{2q, commutes with arbitrary base change due to [SGA4,Exp. XII,Théorème 5.1.iii)]. Hence, the same is true for the character rdet H i pF qs. The assertion now follows from Theorem 4.1.3.b).

A criterion for non-triviality of the twofold covering.
In this subsection, we work with schemes over a base field. We assume that X " P zD, for P non-singular and geometrically irreducible and D Ă P a closed subscheme. The smooth and proper family π : F Ñ X might be extendable over the whole of P , perhaps as a flat, but non-smooth family. The question then naturally arises, whether the twofoldétale covering ̺ π extends over P , or not. In order to investigate this, let us first recall the following well-known fact.
Lemma 4.2.1. Let K be a field, P a non-singular, irreducible K-scheme of finite type, D Ă P a reduced, closed subscheme, and X :" P zD. Denote the irreducible components of D by D 1 , . . . , D m , Z 1 , . . . , Z n , where D 1 , . . . , D m are of codimension 1 and Z 1 , . . . , Z n of codimension ą 1.

{2
that coincides with the one in a).
Proof. a) The scheme P zZ is regular and pD 1 Y . . . Y D m qzZ is a regular, closed subscheme that is purely of codimension 1. Thus, the two schemes form a regular K-pair, so that the assertion follows from [SGA4,Exp. XVI,Corollaire 3.9]. Observe here that Z X X " H. The term on the right hand side would naturally be [SGA4,Exp. XIX,Théorème 3.4]), but we make use of the canonical isomorphism µ 2 -{2 . b) By construction, Z is of codimension at least 2 in P . The terms on the left therefore coincide, due to [SGA1,Exp. X,Corollaire 3.3], while those in the middle obviously agree. Concerning the terms on the right, we note that irreducible and connected components are the same for regular schemes.
Remarks 4.2.2. i) For ζ a geometric generic point on X, there is the canonical isomorphism H 1 et pP, {2 q " Hompπé t 1 pP, ζq, {2 q and analogously for X [SGA4,Exp. VII,Sec. 2]. Thus, to the extendability of a twofoldétale covering, there is an obstruction at every irreducible component of D that is of codimension one, and there are no others. ii) Furthermore, the obstructions may be tested at the non-singular locus of D. iii) Moreover, if a component D j remains irreducible after base extension to the algebraically closed field K then the obstruction at D j may be tested after this base extension.
Theorem 4.2.3 (Criterion for non-triviality of the twofold covering). Let K be a number field, P a non-singular, irreducible K-scheme of finite type, D Ă P a closed subscheme, X :" P zD, and π : F Ñ P a proper and flat family of schemes such that F is non-singular and the restriction π| π´1pXq : π´1pXq Ñ X is smooth. Denote the irreducible components of D by D 1 , . . . , D m , Z 1 , . . . , Z n , where the D j are of codimension 1 and the Z j of codimension ą 1. Assume that D 1 is geometrically irreducible. Furthermore, let an even integer i be given and assume the following. For some geometric point z : K Ñ D lying over a regular point of D that belongs to the component D 1 , the fibre F z has exactly k singular points, all of which are ordinary double points. Then a) (Obstruction) If i " dim F η and k is odd then the twofoldétale covering ̺ π : Y Ñ X associated with π and i is obstructed at the divisor D 1 . In particular, ̺ π is a non-trivial covering and its base extension over X K is still non-trivial. b) (Extendability) i) If i ‰ dim F η and π is projective then ̺ π : Y Ñ X is unobstructed at D 1 . ii) If i " dim F η and k is even then ̺ π : Y Ñ X is unobstructed at D 1 .
Proof. All assertions may be tested after base extension to the algebraic closure K. Then z is a regular point on D. Hence, there exists an affine curve C Ă P that meets D in z transversely. Taking a finite number of further points out, if necessary, we can assume that C X D " tzu holds exactly.
There is a commutative diagram of Gysin short exact sequences in which the downward arrows indicate the restriction homomorphisms. In particular, the obstruction at D 1 may be tested after restriction to C.
There are the canonical isomorphisms H 1 et pC, {2 q " Hompπé t 1 pC, ζq, {2 q and H 1 et pCztzu, {2 q " Hompπé t 1 pCztzu, ζq, {2 q, for ζ a geometric generic point on C. Thus, the obstruction to extendability of a twofoldétale covering from Cztzu to C consists in the operation of the "loop" around z P C.
More precisely, this means the following. The point z P C defines a discrete valuation ring A Ă KpCq and, therefore, a morphism Spec A Ñ C having z and the generic point ζ as the images of the two points on Spec A. Put S :" Spec A h , for A h the Henselisation of A. The scheme S is automatically strictly Henselian, since the residue field K is algebraically closed. Denote by ξ P S the generic point, choose a geometric point ξ lying above ξ, and let ι : ξ Ñ C Ă X be the induced morphism. Abhyankar's lemma [SGA1,Exp. X,Lemme 3.6] shows that πé t 1 pξ, ξq " Galpξ{ξq -p . For a twofoldétale covering of Cztzu, given by a homomorphism χ : πé t 1 pCztzu, ζq ÝÑ {2 , the obstruction is the image of a topological generator of πé t 1 pξ, ξq " Galpξ{ξq -p under the composition πé t 1 pξ, ξq ÝÑ πé t 1 pCztzu, ζq χ ÝÑ {2 . For our particularétale covering, by Theorem 4.1.3.a), this means that one has to consider the operation of πé t 1 pξ, ξq on ι˚Λ max pR i π˚É l pi{2q| X q. Proper base change yields that this sheaf is canonically isomorphic to Λ max pR i π 1 É l pi{2q| ξ q, for π 1 : FˆP S Ñ S the base extension of π.
Next we observe that the total space FˆP S is a non-singular-scheme. Indeed, this is a property that is unaffected by the Henselisation. Thus, it suffices to verify that FˆP C " π´1pCq is non-singular. Due to the smoothness of the morphism π, this is clear, except for the points y 1 , . . . , y k , in which the fibre F z is singular.
On the other hand, each of these singularities is, according to our assumptions, a hypersurface singularity. This means that the kernel L j of the Zariski cotangent map π # : TP ,z Ñ TF ,y j is of dimension one. These kernels disappear under restriction to D 1 , as then the total space becomes singular at y 1 , . . . , y k . Therefore, we must have L 1 " . . . " L k " kerpTP ,z Ñ TD 1 ,z q .
In particular, since C meets L 1 transversely, for we have that N X L j " 0. Consequently, TFˆP C,y j -TF ,y j {π # pNq is of the correct dimension dim FˆP C, because of the injectivity of π # | N . Therefore, the operation of πé t scribed by the Picard-Lefschetz formula [SGA7,Exp. XV,Théorème 3.4]. In the present generality, this states the following.
a) and b.ii) We have that i " dim F η is even. Hence, associated with the singularities on the special fibre, there are the vanishing cycles δ 1 , . . . , δ k P H í et pF ξ , É l pi{2qq, which are uniquely determined up to sign. Their self cup products are known to be xδ 1 , δ 1 y " . . . " xδ k , δ k y " p´1q i{2¨2 , which, in particular, yields that neither of the δ j is equal to zero. The vanishing cycles δ 1 , . . . , δ k are perpendicular to each other. Moreover, the operation of an arbitrary σ P πé t 1 pξ, ξq is given by σpδ j q " εpσqδ j and by σpγq " γ for every γ P H í et pF ξ , É l pi{2qq such that xγ, δ 1 y " . . . " xγ, δ k y " 0.

Concrete description of the twofold covering.
Let us now restrict considerations to the case that the base scheme X satisfies slightly more restrictive conditions. Then twofoldétale coverings may be described rather explicitly.
Definition 4.3.1. We say that a scheme P is a Weil scheme if P is integral, separated, Noetherian, and non-singular.
Lemma 4.3.2. Let X :" P zD, for P a Weil scheme such that 2O P ‰ 0 and D Ă P a closed subscheme. Furthermore, let a twofoldétale covering ̺ : Y Ñ X be given.
Then there exist an invertible sheaf D P PicpP q that is divisible by 2 and a global section ∆ P ΓpP, Dq such that div ∆ is a reduced divisor, supp div ∆ Ď D, and the base change ̺r 1 2 s : Y r 1 2 s Ñ Xr 1 2 s is described by the equation w 2 " ∆ .
Proof. The scheme Y is non-singular, since X is. In particular, Y is normal. Hence, Y may be constructed as follows. Above each affine open Spec A Ă X, put Spec B for B the integral closure of A in the quadratic extension field KpY q.
For the function fields, we have KpY q " KpXqp a gq for some g P KpXq " KpP q. As the extension KpY q{KpXq is unramified, g must be of even valuation at every divisor of X. Hence, there exists some Weil divisor E on P such that E 1 :" divpgq`2E has coefficients only 0 and 1, and only 0 at the prime divisors outside D.
Now recall that our assumptions ensure that every effective Weil divisor C defines an invertible sheaf OpCq and a section s P ΓpP, OpCqq such that divpsq " C. Let ∆ P ΓpP, Dq be the tautological section of the invertible sheaf D :" OpE 1 q. Then w 2 " ∆ defines a twofoldétale covering Y 1 of Xr 1 2 s that is birationally equivalent to Y r 1 2 s. The scheme Y 1 is non-singular, in particular normal, as ∆ nowhere vanishes on X. [SGA1, Exp. I, Proposition 10.1] implies that Y 1 " Y r 1 2 s. Corollary 4.3.3. Let K be a number field, O K its ring of integers, and X :" P zD, for P a Weil scheme that is of finite type over O K , and D Ă P a closed subscheme.
Furthermore, let an even integer i be given and π : F Ñ X be a smooth and proper family of schemes. If i ‰ dim F η then suppose π to be projective.
Then there exist an invertible sheaf D " E b2 P PicpP q and a global section ∆ P ΓpP, Dq such that div ∆ is a reduced divisor, supp div ∆ Ď D, and the following is true. a) The base change ̺ π r 1 2 s : Y r 1 2 s Ñ Xr 1 2 s of the twofoldétale covering ̺ π , associated with π and i, is given by w 2 " ∆. b) For a closed point x P X with finite residue field kpxq of characteristic ‰ 2 and any rational section ι of E without zero or pole at x, one has detpFrob : H í et pF x , É l pi{2qq ýq " 1 ðñ ∆pxq{ι 2 pxq P pkpxq˚q 2 .
Proof. Theorem 4.1.3 applies to π and yields a twofoldétale covering ̺ π : Y Ñ X. From this, Lemma 4.3.2 gives the desired section ∆, shows the assertion of a), and that supp div ∆ Ď D. b) In view of a), the equivalence assertion is a direct consequence of that in Theorem 4.1.3.a).
Corollary 4.3.4. In the situation of Corollary 4.3.3, let x P X K be any point that is closed on the generic fibre. Theǹ for any rational section ι of E without zero or pole at x.
Proof. This follows from Corollary 4.3.3.b), together with the fact that the formation of ̺ π commutes with base change under the inclusion txu Ñ X. Cf. Corollary 4.1.5.
Remark 4.3.5. If ̺ π is trivial, i.e. Y is disconnected, then D " O P and ∆ " u for u P OK a unit. Then, independently of the concrete K-rational fibre in the family, the sequence pdetpFrob: H í et ppF x q p , É l pi{2qq ýqq p is always the same.
4.4. An application. The variation of the characters within a family.
Corollary 4.4.1. Let K be a number field, P a non-singular, irreducible, and projective K-scheme, D Ă P a closed subscheme, X :" P zD, and π : F Ñ P a proper and flat morphism such that the restriction π| π´1pXq : π´1pXq Ñ X is a smooth family of K3 surfaces. a) Then there exist a rational function h H 2 on P and a non-empty open subscheme U Ă P where h H 2 has no poles such that ∆ H 2 pF z q p˙"ˆh H 2 pzq pḟ or every z P UpKq and every prime p Ă O K of residue characteristic ‰ 2. b) There exist a rational function h Pic on P and a non-empty open subscheme U Ă P where h Pic has no poles such that ∆ Pic pF z q p˙"ˆh Pic pzq p˙ for every z P UpKq satisfying rk Pic F z " rk Pic F η and every prime p Ă O K of residue characteristic ‰ 2. c) Assume that there are a prime ideal p Ă K of residue characteristic ‰ 2 and two K-rational points z 1 , z 2 P XpKq such that i) both fibres F z 1 and F z 2 are non-singular, ii) the fibre F z 1 has good reduction at p, and iii) the fibre F z 2 has bad reduction at p, of the type that it has a regular, projective model F z 2 over O K,p , the geometric fibre pF z 2 q p of which has exactly one singular point being an ordinary double point.
Proof The É l -subvector space H alg Ă H 2 et pF η , É l p1q is invariant under the operation of Galpη{ηq ։ πé t 1 pX, ηq and, hence, defines a subsheaf H alg Ă R 2 pπ| π´1pXq q˚É l p1q on X that is again twisted-constant. The twisted-constant sheaf Λ max H alg of rank 1 corresponds to a homomorphism πé t 1 pX, ηq Ñ t1,´1u and, therefore, to a twofold etale covering ̺ π,Pic : Y Pic Ñ X. Locally in the Zariski topology, ̺ π,Pic is given by an equation of the form w 2 " h Pic , as required.
As, for K3 surfaces, the specialisation homomorphism Pic F η b É l Ñ Pic F z b É l is always injective and both sides are assumed to be of the same dimension, the character`∆ Pic pFzq˘h as exactly the property stated. c) The assertion about`∆ H 2 pFz 1 q∆ Pic pFz 1 q˘i s Lemma 2.5.4.i), while that about ∆ H 2 pFz 2 q∆ Pic pFz 2 q˘i s Lemma 2.5.4.ii).
Remark 4.4.2. Parts a and b) together show that, as long as rk Pic F η " rk Pic F z , the jump character`∆ H 2 pFzq∆ Pic pFzq˘d epends on the base point z according to a formula of the type p h jump pzq q, for a rational function h jump on X. Under the additional assumption of part c), h jump is not just a product of a constant with a perfect square.
Fibres with trivial jump character therefore correspond to K-rational points on the double cover w 2 " h. The asymptotics of such rational points is subject to assertions such as Manin's conjecture [FMT]. In practice, however, this seems to be hard to set up. Cf. [EJ12a,Section 5].
Definition 5.1.1. Let P be an irreducible scheme and π : F Ñ P a proper and flat family of schemes such that the generic fibre F η is smooth. Then the discriminant locus of π is the subset of all z P X such that the fibre F z is singular.
Definition 5.1.2 (The normalised discriminant). Let P be a Weil scheme that is proper and flat over and π : F Ñ P be a proper and flat family of schemes, the relative dimension dim F η of which is even. Assume that ‚ the discriminant locus of π is D " D 1 Y . . . Y D m or F 2 Y D 1 Y . . . Y D m , for rF 2 s, D 1 , . . . , D m Ă P distinct irreducible components of codimension 1, and ‚ the twofoldétale covering p̺ π q É associated with π and i " dim F η is obstructed at each pD j q É .
Then, Corollary 4.3.3 provides a unique section ∆ such that div ∆ " pD 1 q`. . .`pD m q or pF 2 q`pD 1 q`. . .`pD m q and ̺ π r 1 2 s is given by w 2 " ∆. We call ∆ the normalised discriminant of the family π.
Remark 5.1.3. In the situation of Definition 5.1.2, suppose for simplicity that F 2 D. There are exactly two sections having the divisor pD 1 q`. . .`pD m q, namely ∆ and p´∆q. Both might classically be called the discriminant. Corollary 4.3.3, however, yields a canonical choice of sign. A particular situation, in which our assumptions are satisfied with m " 1, is provided by the complete intersections of fixed multidegree and fixed even dimension. Cf. Subsection 5.2, below.
Lemma 5.1.4 (An application of the twofold covering). Let π : F Ñ P be a family of schemes that fulfils the assumptions of Definition 5.1.2, and ∆ its normalised discriminant. Moreover, let f : P 1 Ñ P É be any morphism of non-singular, irreducible É-schemes of finite type and write divpf # ∆q " ř M j"1 k j C j for the decomposition into irreducible components.
If, in this situation, ̺ π P 1 is unobstructed at C j , for π P 1 : F ÉˆP É P 1 Ñ P 1 the canonical projection and a certain 1 ď j ď M, then k j is even. In particular, k j ě 2.
Proof. The twofold covering ̺ π is given by w 2 " ∆. According to Corollary 4.1.5, this implies that ̺ π P 1 is given by w 2 " f # ∆. The assertion follows.

Some concrete families.
Complete intersections. Fix positive integers n and d 1 , . . . , d c ě 2 for 1 ď c ď n. For K a field, we will consider complete intersections of multidegree pd 1 , . . . , d c q in P n K . For this, put V :" Proj Sym à 1ďiďc H 0 pP n , Opd i qq _ .
Then V is smooth over . In fact, V is isomorphic to a projective space over . For any -algebra A, an A-valued point on V corresponds to a c-tuple ps 1 , . . . , s c q of sections s i P H 0 pP n A , Opd i qq. Furthermore, there exists a closed subscheme F Ă P n ˆS pec V " P n V satisfying the following condition. Under the canonical morphism π : F Ă P n V ։ V , above each A-valued point x P V pAq, the fibre F x is the closed subscheme of P n A , given by s 1 " . . . " s c " 0, for s 1 , . . . , s c the sections corresponding to x. F is the universal intersection of hypersurfaces of degrees d 1 , . . . , d c in P n .
We also observe that F is smooth over of relative dimension dim V É`n´c . Indeed, projecting the other way round one sees that F is a P dim V É´c -bundle over P n .
Lemma 5.2.1. a) There is a closed subscheme D Ă V of codimension 1 such that the fibre F x is non-singular of dimension n´c if and only if x R D. The scheme D is irreducible and its generic fibre D É is geometrically irreducible. The restriction of π to π´1pV zDq is smooth. b) There is a closed subscheme Z Ă D such that dim F x " n´c if and only if x R Z. The restriction of π to π´1pV zZq is flat.
c) There exists a closed point z P pDzZq É Ă V É such that F z has exactly one singular point, which is an ordinary double point.
Proof. b) The generic fibre of π is of dimension n´c. Hence, the points on V , the fibre over which is of dimension n´c, form an open subset [EGA IV, Corollaire (13.1.5)], having the proposed closed subscheme Z as its complement.
Moreover, F is Cohen-Macaulay, as it is a regular scheme. The flatness asserted thus follows from [EGA IV, Proposition (6.1.5)]. Finally, the claim that Z Ă D is obvious, once the existence of D is established. a) We observe that the non-singularity of a fibre is enough to imply its smoothness [EGA IV, Proposition (6.7.7)], since all residue fields occurring are perfect. Therefore, D is closed according to [EGA IV, Corollaire (6.8.7)], together with the properness of π. The final assertion of a) follows from the facts that π| π´1pV zDq is flat and all its fibres are smooth [EGA IV,Corollaire (17.5.2)].
Only the assertions that D is irreducible and of codimension 1 and that D É is geometrically irreducible are particular to our situation. Irreducibility of D É is proven in [Ben12,Lemma 7.3.2], as well as irreducibility of the intersection of D with every special fibre V p of V . These properties together are sufficient for irreducibility of D. Furthermore, the fact that D is of codimension 1 follows from [Ben12,Corollaire 7.3.3 and Lemme 7.4.4.i)]. c) This assertion, finally, is established in [Ben12,Lemme 7.4.3.ii)].

Let
S be a non-singular complete intersection of multidegree pd 1 , . . . , d c q in P n L , for L a number field. Then there exists a unique L-rational point x P pV zDqpLq such that F x " S. Moreover, i) let 0 ď i ď 2pn´cq but i ‰ n´c. If i is odd then H í et pF x , É l q " 0. On the other hand, if i is even then dim H í et pF x , É l q " 1 by the weak Lefschetz theorem [SGA4, Exp. XIV, Corollaire 3.3 and Exp. XVI, Corollaire 3.9]. Moreover, Frob operates identically, hence det Frob " 1. ii) For the middle cohomology, one has N :" dim H n´ć et pF x , É l q " p´1q n´c " d 1¨¨¨dc ÿ e 0`¨¨¨`ec "n´c pe 0`1 qp1´d 1 q e 1¨¨¨p 1´d c q ec´n`c`p´1q n´c´1 2 ı by [Hi,Satz 2.4]. Concerning the operation of Frob, there is the theorem below.
Theorem 5.2.3. Let i " n´c be even. Then the normalised discriminant ∆ is a section ∆ P O V pDq such that div ∆ " pDq. It has the property below. Let K be a number field and x P pV zDqpKq be any K-rational point. Then, for any prime p Ă O K of good reduction and residue characteristic ‰ 2, detpFrob: H n´ć et ppF x q p , É l p n´c 2 qq ýq "ˆ∆ pxq p˙. Proof. From Corollary 4.3.3, we know that supp div ∆ Ď D and that div ∆ is reduced. Thus, in view of the irreducibility of D, the only options are div ∆ " 0 and div ∆ " pDq. Moreover, since there are fibres over points on D with exactly one ordinary double point, Theorem 4.2.3.a) excludes the first option.
For the second assertion, note that, since V is proper over , x extends uniquely to an O K -valued point x : Spec O K Ñ V . The fibre product FˆP ,x Spec O K is the canonical model over O K of the complete intersection F x . The reduction pF x q p is nothing but the fibre π´1pxppqq. In particular, the primes of good reduction of F x are precisely those, for which xppq P V zD. The assertion is therefore a special case of Corollary 4.3.3.b).
Remark 5.2.4. The particular case of Theorem 5.2.3 for cubic surfaces had been obtained before [EJ12a,Theorem 2.12] in a project on constructing cubic surfaces with particular Galois actions on the 27 lines.
A particular case. Hypersurfaces.
Remark 5.2.5. In the special case of hypersurfaces of degree d, i.e. the case that c " 1, the parameter space V coincides with the Hilbert scheme. A number of further simplifications occur. i) Most notably, the subscheme Z as in Lemma 5.2.1 is empty.
ii) An explicit example of a hypersurface of degree d over É with exactly one singular point, being an ordinary double point, is provided by the equation iii) Moreover, the irreducibility of D may be directly seen as follows. Consider the d-uple embedding ι : P n ãÑ P M , for M :"`n`d n˘´1 . Then the points on D correspond to the hyperplanes in P M that meet ιpP n q somewhere tangentially. Hence, D is the image of the tangent bundle T ιpP n q under a morphism of schemes. As T ιpP n q itself is irreducible, the claim follows. iv) Finally, it is known since the days of G. Boole ([Bo1,p. 19] and [Bo2,p. 171]) that deg D " pd´1q n pn`1q. In particular, one explicitly sees that deg D is always even in the case of even-dimensional hypersurfaces. A present-day argument for this goes roughly as follows. Let F 0 and F 1 be two homogeneous forms of degree d defining smooth hypersurfaces. One is interested in the number of all pµ : νq P P 1 such that X pµ:νq :" V pµF 1`ν F 2 q is singular.
Let us count the number of singular points instead. According to the Euler identity, x P X pµ:νq is singular if and only if µ BF 0 BX j pxq`ν BF 1 BX j pxq " 0, for j " 0, . . . , n. The question is thus, for how many points x P P n one has ϕ 0 pxq " ϕ 1 pxq, where ϕ 0 and ϕ 1 are the morphisms ϕ i : P n ÝÑ P n , x Þ Ñ p BF i BX 0 pxq :¨¨¨: BF i BXn pxqq . This means to calculate the intersection number of the corresponding two graphs. As rΓ 0 s " rΓ 1 s " n ř k"0 rH n´kˆp pd´1qHq k s P CH n pP nˆPn q , the claim that rΓ 0 s¨rΓ 1 s " pd´1q n pn`1q easily follows.
Remark 5.2.6. In the particular case of a hypersurface, the normalised discriminant provided by Theorem 5.2.3, is essentially due to T. Saito [Sa]. M. Demazure's divided discriminant [Dem,Section 5] differs from the normalised discriminant by a factor of p´1q d´1 2 , when d is odd, and p´1q d 2 n`2 2 , when d is even [Sa,Theorem 3.5]. In our arguments below, we will make use of Demazure's calculations of discriminants of particular hypersurfaces.
Double covers. 1 Fix positive integers n and d, where d is even. For K a number field, we will consider double covers of P n K ramified at a degree d hypersurface. For this, put W :" Proj Symp ' H 0 pP n , Opdqq _ q. Then W is smooth over , since W is isomorphic to a projective space over . For any -algebra A, an A-valued point on W corresponds to a pair pt, sq, where t P A and s P H 0 pP n A , Opdqq. Furthermore, there exists a closed subscheme F Ă P n ˆS pec W " P n W satisfying the following condition. Under the canonical morphism π : F Ă P n W ։ W , above each A-valued point x P V pAq, the fibre F x is the double cover of P n A , given by tw 2 " s , for pt, sq the pair corresponding to x. F is the universal double cover of P n ramified at a degree d hypersurface.
Projecting the other way round, one sees that F is a P dim W É´1 -bundle and, in particular, smooth over the weighted projective space Ppd{2, 1 n q .
Lemma 5.2.7. The closed subset D Ă W parametrising singular double covers is the union of three irreducible components. These are the cone C D d over the locus D d Ă V d " V parametrising singular hypersurfaces of degree d, the hyperplane H 0 corresponding to the case t " 0, and the special fibre W 2 .
Proof. The three components correspond to the three ways a double cover may become singular. 5.2.8. Let S be a double cover of P n L ramified at a non-singular hypersurface of degree d, for L a number field. Then there exists a unique L-rational point x P pW zDqpLq such that F x " S. Furthermore, i) let 0 ď i ď 2n but i ‰ n. Then H í et pF x , É l q " 0 for i odd and dim H í et pF x , É l q " 1 for i even. Indeed, weighted projective spaces have the same cohomology groups as ordinary ones [Ka,Theorem 1], so that the assertions are consequences of the weak Lefschetz theorem. Moreover, in the case that i is even, Frob operates identically, hence det Frob " 1. ii) On the other hand, as follows from [Hi,Satz 2.4] by a simple calculation. Concerning the operation of Frob, there is the theorem below.
Theorem 5.2.9. Let i " n be even. Then the normalised discriminant ∆ is a section ∆ P O W pDq such that div ∆ " pC D d q`pH 0 q. It has the property below. Let K be a number field and x P pW zpC D d YH 0 qqpKq be any K-rational point. Then, for any prime p Ă O K of good reduction and residue characteristic ‰ 2, detpFrob: H ń et ppF x q p , É l pn{2qq ýq "ˆ∆ pxq p˙.
Remarks 5.2.10. i) The result says, in particular, that the relationship between the normalised discriminant of the double cover tw 2 " s and the divided discriminant [Dem,Définition 4] of the homogeneous form s of degree d is given by the formula ∆pt, sq "˘t∆ d psq .
ii) Moreover, the sign is p´1q nd 4 . Recall that n and d are assumed to be even. Proof. Indeed, for theétale Euler characteristic, formula (13) implies that On the other hand, for s 0 :"´X d 0´. . .´X d n , the double cover has no real points. In this situation, [Sa,Corollary 1.3.1] shows that ∆p1, s 0 q has the same sign as " p´1q if d " 0 pmod 4q , p´1q n 2`1 if d " 2 pmod 4q . But ∆ d ps 0 q ă 0, according to [Dem, Section 5, Exemple 1].
5.2.11. Proof of Theorem 5.2.9. From Corollary 4.3.3, it is known that div ∆ is reduced and that supp div ∆ Ď C D d Y H 0 Y W 2 . Thus, for div ∆, there are in principle eight options.
In order to decide which one actually occurs, we first note that there exists a closed point x P C D d Ă W such that F x has exactly one singular point, which is an ordinary double point. A corresponding double cover is given by Thus, Theorem 4.2.3 applies and shows that supp div ∆ Ě C D d . Moreover, is odd, since n and d are even. This excludes the option supppdiv ∆q É " pC D d q É , since projective spaces do not allow double covers ramified at an odd degree hypersurface. Consequently, supppdiv ∆q É " pC D d YH 0 q É .
The non-occurrence of W 2 in div ∆ is slightly more subtle. In order to see it, let us first consider the more complicated double cover of Spec , given by . .`X n´1 X d´1 n q " 0 . This scheme is smooth over 2, therefore the double cover ̺ πv : Y Ñ Spec associated with it is unramified above the prime 2. Moreover, over Spec r 1 2 s, the scheme π v is isomorphic to the double cover π u : w 2 " s 0 , for Consequently, ̺ πu has the same monodromy around 2 as ̺ πv , which means that it is unramified at 2, too. In order to complete the proof of the first assertion, we need to verify that ∆ d ps 0 q is of even 2-adic valuation. For this, we first note that ∆ d pX d 0`X 0 X d´1 1`X 1 X d´1 2`. . .`X n´1 X d´1 n q is odd [Dem,Lemme 10]. From the auxiliary form chosen, s 0 is obtained by applying the matrix diag`1, 4 which is of determinant 4 n d`1 d 2 r1`p´1 q n´1 pd´1q n s . Hence, [Dem,Proposition 11.e)] shows that ν 2 p∆ d psqq " 2rnpd´1q n`p d´1q n`p´1 q n´1 d s " 0 pmod 2q. For the second assertion, let us first note that, since W is proper over , x extends uniquely to a O K -valued point x : Spec O K Ñ W . The reduction pF x q p is nothing but the fibre π´1pxppqq. In particular, the primes of good reduction of F x are precisely those, for which xppq P W zpC D d YH 0 YW 2 q. Thus, the assertion is a special case of Corollary 4.3.3.
Examples 5.2.12. i) For the case that n " 2 and d " 4, i.e. that of degree two del Pezzo surfaces, one may particularly consider the blow-up of P 2 in seven É-rational points in general position. It is a standard procedure to rewrite such a blow-up in the form w 2 " s, for s a quartic in three variables.
Then ∆p1, sq is clearly a square in É˚, as the Galois operation on H 2 et ppF 1,s q É , É l p1qq is trivial. On the other hand, the second author's code for calculating the Dixmier-Ohno invariants [El] immediately shows that ∆ d psq is a square, too.
ii) For the case that n " 2 and d " 6, i.e. that of degree two K3 surfaces, an experiment immediately shows that ∆p1, x 6 0`x 6 1`x 6 2 q is minus a perfect square, cf. Example 2.6.7. On the other hand, one has ∆ d px 6 0`x 6 1`x 6 2 q " 2 54 3 54 . 5.3. Special space quartics.
The discriminant of special quartics. The Hilbert scheme of quartics in P 3 is a 34-dimensional projective space P . According to Boole's formula, the normalised discriminant is a section ∆ P ΓpP, Op108qq. In other words, ∆ is a polynomial of degree 108 in the 35 coefficients. In the case of space quartics, the divided discriminant coincides with the normalised discriminant.
As it seems impossible to work with ∆ experimentally, at least with current technology, we restrict our considerations to special quartics. By which we mean those that are given by an equation of the form cX 4 3`f 2 pX 0 , X 1 , X 2 qX 2 3`f 4 pX 0 , X 1 , X 2 q " 0 . The Hilbert scheme of special quartics is a 21-dimensional linear subspace P 1 Ă P . It turns out that the restriction of ∆ to special quartics may be expressed entirely in terms of the divided discriminant of plane quartics, which we denote by δ and which is homogeneous of degree 27.
Proposition 5.3.1. a) The polynomial map pc, f 2 , f 4 q Þ Ñ δpf 2 2´4 cf 4 q splits off a factor of c 14 . Moreover, the cofactor, which we denote by δ dP , is irreducible. b) The restriction of ∆ to special quartics factors into 2´5 2 cδpf 4 qδ 2 dP . Proof. According to Lemma 5.3.2, the splitting of δpf 2 2´4 cf 4 q into irreducible factors has the form c m δ dP . Moreover, an experiment immediately shows that m ď 14. We used the code described in [El] on concrete values for f 2 and f 4 , leaving c an indeterminate.
On the other hand, a special quartic may be given by the system of equations cw 2`f 2 pX 0 , X 1 , X 2 qw`f 4 pX 0 , X 1 , X 2 q " 0 , X 2 3 " w , which generically defines a double cover of a degree two del Pezzo surface, ramified at the curve given by w " 0. The latter is isomorphic to the plane curve V pf 4 q.
Thus, there are only two ways, a special quartic may become singular. Either the double cover defined by equation (14) already is or the ramification locus is singular. The latter means δpf 4 q " 0, the former δpf 2 2´4 cf 4 q " 0, i.e. c " 0 or δ dP " 0. Consequently, the restriction of ∆ to special quartics splits into exactly these three irreducible factors.
Next, we consider a general point x P V pδ dP q. The corresponding fibre F x is a double cover, ramified at a non-singular curve, of a weak del Pezzo surface of degree 2 having exactly one singularity, which is an ordinary double point. Thus, on F x , there are precisely two ordinary double points. Theorem 4.2.3.b.ii) shows that the associated double cover ̺ : Y Ñ P 1 does not ramify at V pδ dP q. According to Lemma 5.1.4, the discriminant must contain the factor δ dP with an even exponent. Therefore, the discriminant is divisible by cδpf 4 qδ 2 dP , which is homogeneous of degree 1`27`2p54´mq " 136´2m. As the total degree must be equal to 108, this implies both, that m is exactly equal to 14 and that the discriminant coincides with cδpf 4 qδ 2 dP , up to constant factor. Finally, the facts that the diagonal surface given by X 4 0`X 4 1`X 4 2`X 4 3 " 0 is of discriminant 2 176 and that δpX 4 0`X 4 1`X 4 2 q " 2 40 reveal that this factor is equal to 2´5 2 . Proof. Assume to the contrary, that F contains two irreducible factors E 1 and E 2 , possibly equal to each other, neither of which is just associated to c.
Then both, E 1 and E 2 , effectively depend on coefficients of f d . Indeed, if for example E 1 would be a polynomial only in c and the coefficients of f d{2 then the Hilbert Nullstellensatz would yield a polynomial g P ÉrX 0 , . . . , X n s d 2 such that ÉrX 0 , . . . , X n s d Ñ É, f d Þ Ñ E 1 p1, g, f d q is the zero map. But then f d Þ Ñ F pg 2´4 f d q would be the zero map, as well, a contradiction.
Thus, there exists some polynomial h P ÉrX 0 , . . . , X n s d{2 such that both polynomial functions f d Þ Ñ E 1 p1, h, f d q and f d Þ Ñ E 2 p1, h, f d q are non-constant. But this means that ÉrX 0 , . . . , X n s d Ñ É, f d Þ Ñ F ph 2´4 f d q is reducible, in contradiction with the irreducibility of F .
The algebraic part of the discriminant in the case of special quartics.
Proof. The surface S comes equipped with a finite morphism p : S Ñ S 1 , which is generically 2 : 1, to the underlying degree two del Pezzo surface S 1 . The induced homomorphism p˚: Pic S 1 K Ñ Pic S K doubles all intersection numbers. As, on a degree two del Pezzo surface, there are no non-trivial invertible sheaves that are numerically equivalent to zero, we see that p˚is necessarily injective. The assertion follows.
Suppose that rk Pic S K " 8, exactly. Then Pic S K perhaps differs from p˚Pic S 1 K by some finite index, but the algebraic part of the discriminant depends only on the Picard representation PicpS K q b É. Hence ∆ Pic pSq " ∆ Pic pS 1 q .
Moreover, for del Pezzo surfaces, the Chern class homomorphism is surjective. Hence, ∆ Pic pS 1 q coincides, up to squares, with the normalised discriminant of S 1 . However, S 1 is isomorphic to the double cover S 2 : w 2 " f 2 2´4 cf 4 . Thus, according to Remark 5.2.10, the latter is equal to δpf 2 2´4 cf 4 q. We have just shown part a) of the lemma below.
Lemma 5.3.5. Let K be a number field and S a K3 surface over K, given by a special quartic cX 4 3`f 2 pX 0 , X 1 , X 2 qX 2 3`f 4 pX 0 , X 1 , X 2 q " 0 . Assume that S is of geometric Picard rank 8. a) Then the algebraic part of the discriminant is δpf 2 2´4 cf 4 q. b) The jump character is´c Concrete examples.
Example 5.3.6 (cf. Example 2.6.11). Let S be the K3 surface over É, given by equation (10). Then the geometric Picard rank of S is 8 and the jump character of S is trivial.
Remark 5.3.7. This example, and several others of the same kind, were found by a systematic inspection of all special quartics with coefficients from t´1, 0, 1u.
We organised the computations as follows. First, we determined all ternary quartic forms with coefficients from t´1, 0, 1u, up to the equivalence provided by permutation and sign change of the variables. This led to a list of 605 394 forms, 511 308 of which are smooth. We then equipped each of these quartics with its individual list of potential quadrics with coefficients from t´1, 0, 1u. We always made use of the fact that f 2 and p´f 2 q result in isomorphic surfaces. Moreover, we used the additional symmetry, whenever f 4 was stable under one of the transformations above. This resulted in a total of 183 098 318 smooth K3 surfaces. Out of these, for just 709 the product δpf 4 qδpf 2 2´4 f 4 q is a square. For these, we therefore showed that they have a trivial jump character, as soon as they have geometric Picard rank 8. The computation using magma took 330 hours on one core of an Intel(R) Core(TM)i7-3770 processor running at 3.40GHz.
Example 5.3.8 (cf. Example 2.6.12). Let S be the K3 surface over É, given by equation (11). Then the geometric Picard rank of S is 8 and the jump character of S is p´1 . q.
Remark 5.3.9. The same argument shows that the jump character is p´1 . q for every K3 surface of the type X 4 3`f 4 pX 0 , X 1 , X 2 q and geometric Picard rank 8.