A note on Seshadri constants of line bundles on hyperelliptic surfaces

We study Seshadri constants of ample line bundles on hyperelliptic surfaces. We obtain new lower bounds and compute the exact values of Seshadri constants in some cases. Our approach uses results of F. Serrano (1990), B. Harboune and J. Roe (2008), F. Bastianelli (2009), A.L. Knutsen, W. Syzdek and T. Szemberg (2009).


Introduction
Seshadri constants measure how positive a line bundle is. They were introduced in 1992 by J.P. Demailly in [De1992] as an attempt to tackle the famous Fujita conjecture. The conjecture has not been proven but Seshadri constants soon became an object of study on their own.
Giving exact values or just estimating Seshadri constants is very hard, even in case of line bundles on algebraic surfaces, see e.g. [Ba1999]. There exists an upper bound for Seshadri constant of a line bundle at points x 1 , . . ., x r on a smooth projective n-dimensional variety X, namely ε(L, x 1 , . . . , x r ) ≤ n L n r . Therefore it is interesting to look for lower bounds. There are several results concerning Seshadri constants on surfaces with Kodaira dimension zero. Let us recall some of them. In appendix to [Ba1998], Th. Bauer and T. Szemberg give upper bound for the global Seshadri constant of an ample line bundle on an abelian surface and as a corollary obtain that the Seshadri constant of such a line bundle is always rational. In [Ba1997] Th. Bauer computes Seshadri constants on all K3 surfaces of degree 4. This result is extended by C. Galati and A.L. Knutsen in [GaK2013] who compute Seshadri constants on K3 surfaces of degrees 6 and 8. Earlier in [K2008] A.L. Knutsen estimates Seshadri constants on K3 surfaces with Picard number 1. T. Szemberg in [Sz2001] proves that the global Seshadri constants on Enriques surfaces are always rational and also provides the lower bound for Seshadri constant at an arbitrary point. Up to our knowledge, Seshadri constants have not been studied on hyperelliptic surfaces before.
We estimate Seshadri constants on hyperelliptic surfaces, in some cases we compute their exact values. The paper is organised in the following way -in Theorem 3.1 we compute the global Seshadri constant of a line bundle of type (1,1) on a hyperelliptic surface of an arbitrary type, next in Proposition 3.3 we point out a hyperelliptic surface type and a point at which a Seshadri constant of a line bundle of type (1,1) is strictly greater than 1. In Theorem 3.4 we compute the global Seshadri constant of a arbitrary ample line bundle on hyperelliptic surface of type 1, and in Theorem 3.5 we provide a lower bound for this constant on hyperelliptic surfaces of types 2-7. Finally, in Theorem 3.6 we estimate from below the multi-point Seshadri constant of an ample line at r very general points on hyperelliptic surfaces.

Notation and auxiliary results
Let us set up the notation and basic definitions. We work over the field of complex numbers C. We consider only smooth reduced and irreducible projective varieties. By D 1 ≡ D 2 we denote the numerical equivalence of divisors D 1 and D 2 . By a curve we understand an irreducible subvariety of dimension 1. In the notation we follow [Laz2004].
Let X be a smooth projective variety and L a nef line bundle on X. We recall the definition of a Seshadri constant.
Definition 2.1. (1) The Seshadri constant of L at a given point x ∈ X is the real number where the infimum is taken over all irreducible curves C ⊂ X passing through x.
(2) The global Seshadri constant of L is defined to be Let x 1 , . . ., x r be pairwise distinct points. The notion of a Seshadri constant of a line bundle at a point may be generalised to r points in the following way: Definition 2.2. The multi-point Seshadri constant of L at x 1 , . . ., x r is the real number where the infimum is taken over all irreducible curves C ⊂ X passing through at least one of the points x 1 , . . ., x r .
For a fixed line bundle L the function (x 1 , . . . , x r ) → ε(L, x 1 , . . . , x r ) is constant for points in very general position, moreover its value for points not lying in very general position does not exceed the value for points in very general position -see [Laz2004], Example 5.1.11. We denote the Seshadri constant of L at r points in very general position by ε(L, r).
Let (2) for every m ∈ N such that 1 ≤ m < µ r−1 and if for every k ∈ Z such that k 2 < r r−1 min{m, m + k} we have For more background on Seshadri constants we refer to [PSC2009].
Now let us recall the definition of a hyperelliptic surface.
Definition 2.4. A hyperelliptic surface S (sometimes called bielliptic) is a surface with Kodaira dimension equal to 0 and irregularity q(S) = 1.
where A and B are elliptic curves, and G is an abelian group acting on A by translation and acting on B, such that A/G is an elliptic curve and B/G ∼ = P 1 ; G acts on A × B coordinatewise. Hence we have the following situation: where Φ and Ψ are natural projections.
Hyperelliptic surfaces were classified at the beginning of 20th century by G. Bagnera and M. de Franchis in [BF1907], and independently by F. Enriques i F. Severi in [ES1909-10]. They showed that there are seven non-isomorphic types of hyperelliptic surfaces. Those types are characterised by the action of G on B ∼ = C/(Zω ⊕ Z) (for details see eg. [Bea1996], VI.20). For every hyperelliptic surface we have that the canonical divisor K S is numerically trivial.
In 1990 F. Serrano in [Se1990] characterised the group Num(S) for each of the surface's type: Theorem 2.5 (Serrano). A basis of the group of classes of numerically equivalent divisors Num(S) for each of the surface's type and the multiplicities of the singular fibres in each case are the following: Let µ = lcm{m 1 , . . . , m s } and let γ = |G|. Given a hyperelliptic surface, its basis of Num(S) consists of divisors A/µ and (µ/γ) B. We say that L is a line bundle of type (a, b) on a hyperelliptic surface if L ≡ a · A/µ + b · (µ/γ)B. In Num(S) we have that A 2 = 0, [Ap1998], Proposition 5.2).
The following proposition holds: Proposition 2.6 (see [Se1990], Lemma 1.3). Let D be a divisor of type (a, b) on a hyperelliptic surface S. Then (1) χ(D) = ab; (2) D is ample if and only if a > 0 and b > 0; Now we recall a bound for the self-intersection of a curve. Adjunction formula, applied to the normalisation of a curve C, implies the following formula: Remark 2.7 (Genus formula, [GH1978], Lemma, p. 505). Let C be a curve on a surface S, passing through x 1 , . . ., x r with multiplicities respectively m 1 , . . ., m r . Let g(C) denote the genus of the normalisation of C. Then Note that a curve C on a hyperelliptic surface has genus at least 1. Otherwise the normalisation of C, of genus zero, would be a covering (via Φ) of an elliptic curve A/G. This contradicts the Riemann-Hurwitz formula.
For families of curves we have Xu-type lemma. The original version of this lemma was proved by G. Xu in [Xu1995]. We will use the generalisation of the Xu Lemma obtained by A.L. Knutsen, W. Syzdek, T. Szemberg in [KSSz2009], and independently by F. Bastianelli in [Bas2009]. Let gon(C) denote the gonality of a smooth curve C, i.e. the minimal degree of a covering C → P 1 .
Applying the Xu-type lemma to a family C of curves passing through x 1 , . . ., x r with multiplicities respectively m 1 , . . ., m r , where m 1 ≥ 2, on a blow-up at x 2 , . . ., x r , we have the following multi-point version of the Xu-type lemma Lemma 2.9. For a general curve C of the family C as above we have that Every hyperelliptic surface S is nonrational, hence for every curve C ⊂ S we have gon( C) ≥ 2 (see [KSSz2009], remarks following Theorem A).

Main results
3.1. Seshadri constants of ample line bundles on hyperelliptic surfaces. We start with computing the global Seshadri constant in the simplest case of an ample line bundle on a hyperelliptic surface, i.e. for a line bundle of type (1, 1).
Theorem 3.1. Let S be a hyperelliptic surface. Let L be a line bundle of type (1, 1) on S. Then ε(L) = 1.
Proof. Let C ≡ (α, β) denote an irreducible curve passing through a given point x ∈ S with multiplicity m, m ≥ 1. We estimate the value of LC m from below. Depending on the position of a point x and on hyperelliptic surface's type, we have the following possibilities for C to be an irreducible curve: (1) C ≡ B ≡ (0, k) and x is an arbitrary point, where k = 1 for a hyperelliptic surface of an odd type; k = 2 for a hyperelliptic surface of type 2 and 4; k = 3 for a hyperelliptic surface of type 6. Then LC m = k 1 ≥ 1.
By the proof of Theorem 3.1 we immediately obtain a corollary: On the other hand, it is not true that on every hyperelliptic surface the equality ε(L, x) = 1 holds for every x ∈ S. Proposition 3.3. There exists a hyperelliptic surface S such that for a line bundle L of type (1, 1) Proof. Let S be a hyperelliptic surface of type 2, and let L be a line bundle of type (1, 1) on S. Let x be a very general point on S. We will prove that ε(L, x) ≥ 4 3 . Let C ≡ (α, β) be an irreducible curve passing through a given point x ∈ S with multiplicity m, m ≥ 1.
Using the same method as presented in Theorem 3.3 one can show that for a very general point x on a hyperelliptic surface of type 2 and for L of type (1, 1), the Seshadri constant of L at x is greater than a constant slightly bigger than 4 3 . The proof splits in a large number of cases and therefore we decide not to present it here. However precise study of this example might support the idea that this Seshadri constant is irrational. Now we will prove a lower bound for the global Seshadri constant of an arbitrary ample line bundle on hyperelliptic surface of type 1. Proof. Let S be a hyperelliptic surface of type 1, let L ≡ (a, b). Let C ≡ (α, β) denote an irreducible curve passing through a given point x with multiplicity m, m ≥ 1. Using Bézout's Theorem we obtain: if C ≡ A/2 and x lies on the singular fibre A/2; 2b, if C ≡ A and x lies on the fibre A; a + b, if C ≡ (α, β) and x lies on one of the singular fibres A/2; a 2 + b, if C ≡ (α, β) and x lies on one of the fibres A. Hence on a hyperelliptic surface of type 1 ε(L) = min{a, b}.
By the theorem above we see that on a hyperelliptic surface of type 1 the global Seshadri constant of an ample line bundle L is always submaximal, ie. smaller than √ L 2 . Note that the method used in Theorem 3.4 does not work on hyperelliptic surfaces of other types. For hyperelliptic surfaces of type 1 the lower bound of LC m , where a curve C is not a fibre, is always greater than the value of LC m for some fibre C. It is also easy to show for which fibre and for which point position the global Seshadri constant is actually reached. This is not the case for hyperelliptic surfaces of types 2-7.
For hyperelliptic surfaces of types 2-7, we have the following lower bound for the global Seshadri constant Proof. We have that L ≡ (a, b) ≡ min{a, b} · M + N, where M ≡ (1, 1) and N is nef. By definition of a Seshadri constant, for every x ∈ S ε(L, x) ≥ min{a, b} · ε(M, x) + ε(N, x) ≥ min{a, b} · ε(M, x).

Multi-point Seshadri constants of ample line bundles on non-rational surfaces.
In this section we present a lower bound for Seshadri constant at r points in very general position on hyperelliptic surfaces. The lower bound for multi-point Seshadri constants obtained in Theorem 3.6 is not far from the upper bound. As mentioned before, it is well-known (see e.g. [PSC2009], Proposition 2.1.1) that for smooth projective surfaces ε(L, r) ≤ L 2 r .
Biran-Nagata-Szemberg conjecture says that for any algebraic surface there exists r 0 > 0 such that for every r > r 0 in fact there is an equality ε(L, r) = L 2 r .
Theorem 3.6. Let S be a hyperelliptic surfaces. Let L be an ample line bundle on S. Then Proof. The claim follows immediately from Harbourne-Roé theorem (Theorem 2.3) with µ = 8. The point is to check that the assumptions of this theorem are satisfied with that particular constant. Turning into details we need to check the following two conditions: (1) for every integer 1 ≤ m < 8 (2) for every integer 1 ≤ m < 8 r−1 and for every integer k with k 2 < r r−1 min{m, m + k} Ad.
As L is ample, by Hogde Index Theorem it is enough to prove that We split the proof that C 2 ≥ m 2 r − 1 8 into two cases: m = 1 and m > 1. For m = 1, we have h 0 (C) = dim |C| + 1 ≥ r + 1. Moreover by Proposition 2.6 (3), h 0 (C) = C 2 2 . Hence C 2 2 ≥ r + 1. Therefore it is enough to show that This condition is satisfied for every positive r.
Simple computations confirm that the last inequality is satisfied for all admissible m > 1, r and k. The proof is completed.
Remark 3.7. Note that the Theorem 3.6 holds also for abelian surfaces with ρ = 1.