Anticanonical geometry of the blow-up of $\mathbb{P}^4$ in $8$ points and its Fano model

Building on the work of Casagrande-Codogni-Fanelli, we develop our study on the birational geometry of the Fano fourfold $Y=M_{S,-K_S}$ which is the moduli space of semi-stable rank-two torsion-free sheaves with $c_1=-K_S$ and $c_2=2$ on a polarised degree-one del Pezzo surface $(S,-K_S)$. Based on the relation between $Y$ and the blow-up of $\mathbb{P}^4$ in $8$ points, we describe completely the base scheme of the anticanonical system $|{-}K_Y|$. We also prove that the Bertini involution $\iota_Y$ of $Y$, induced by the Bertini involution $\iota_S$ of $S$, preserves every member in $|{-}K_Y|$. In particular, we establish the relation between $\iota_Y$ and the anticanonical map of $Y$, and we describe the action of $\iota_Y$ by analogy with the action of $\iota_S$ on $S$.

In dimension 4, toric Fano manifolds have been classified by Batyrev [Bat99] and Sato [Sat00], and Fano manifolds of index r > 2 have been classified. Among Fano fourfolds, those of index one have special positions: Küchle constructed a number of examples with Picard number one, and explained some known results with lists of related problems (see [Küc97]). To find and classify Fano fourfolds of index one, Coates, Corti and others have embarked on a program using mirror symmetry ( [CCG + 13], and see a list of examples in [CGKS20]), where heavy computer calculations are involved. A complete classification might not be desirable, but it is interesting to exhibit some Fano fourfolds with special geometric properties, for example, those with Picard number ρ close to the conjectural boundary ρ ≤ 18, or those whose anticanonical system has non-empty base locus. In order to study Fano manifolds with large Picard number (see [Cas12]), Casagrande introduced the invariant called Lefschetz defect, and developed fruitful results in this direction ( [Cas13,CR22]).
Let Y := M S,−K S be the moduli spaces of semi-stable rank-two torsion-free sheaves with c 1 = −K S , c 2 = 2 on a polarised degree-one del Pezzo surface (S, −K S ). The moduli spaces Y = M S,−K S form a remarkable family of smooth Fano fourfolds with Picard number 9. The study of this family is motivated by two issues. Firstly, for Fano fourfolds with large Picard number (e.g. at least 7), only few examples which are not products of del Pezzo surfaces are known. As pointed out in [CCF19, Sect. 1,B], the family of Fano fourfolds Y is the only known example of Fano fourfolds with Picard number at least 9, which is not a product of surfaces. Secondly, it is delicate to find examples of Fano fourfolds whose anticanonical system has non-empty base locus, since most Fano fourfolds classified so far are toric, which implies that any ample line bundle on them is globally generated. Some examples are constructed in [Heu16,Chapter 6.3] as complete intersections of two hypersurfaces in weighted projective spaces; two families are identified in [Sec21] as Fano fourfolds with Picard number 3 and having some contraction onto a smooth Fano threefold. In [CCF19, Thm. 1.10], it is shown that the base locus of the anticanonical system |−K Y | has positive dimension. Therefore, the geometry of Y is worth detailed understanding.
A. The anticanonical system of the Fano model Y . The birational geometry of Y = M S,−K S is related to the birational geometry of the blow-up X of P 4 at 8 points. In [CCF19,Lem. 5.18], an explicit relation between X and Y is given: the Fano fourfold Y is obtained from X by flipping the strict transforms of the lines through all pairs of blown up points and of the quartic curves through 7 blown up points in P 4 . Thanks to this relation between X and Y , it is shown in [CCF19, Lem. 7.5, Cor. 7.6] that the base locus of |−K Y | contains the strict transform R Y of a smooth rational quintic curve passing through the 8 blown up points in P 4 , and that |−2K Y | is base-point-free. We complete the study of the anticanonical system and show more precisely that: Theorem 1.1. For the Fano fourfold Y := M S,−K S , the base scheme of |−K Y | is the reduced smooth curve R Y .
As a direct application, we obtain the smoothness of a general member in the anticanonical system. Corollary 1.2. Let D ∈ |−K Y | be a general divisor. Then D is smooth.
B. The Bertini involution of the Fano model Y . Now we turn our attention to the automorphism group of Y . In [CCF19,Sect. 4], a group morphism ρ between the Picard groups of the degree-one del Pezzo surface S and of the moduli spaces Y = M S,−K S is defined. This morphism ρ induces an isomorphism between the automorphism groups of S and of Y (see [CCF19,6.15]). In particular, there is an involution ι Y of Y which is induced by the Bertini involution ι S of S.
We mention here that another motivation behind the study of the Bertini involution ι Y is the understanding of the corresponding birational involutions ι X of X and ι P 4 of P 4 . These birational maps ι X and ι P 4 are classically known, as they can be defined via the Cremona action of the Weyl group W (E 8 ) on sets of 8 points in P 4 (see [DO88] and [DV81]). Nevertheless, the classical definitions of ι X and ι P 4 do not give a geometric description of these maps. In [CCF19, Prop. 8.9, Cor. 8.10], a factorisation of these maps is given as smooth blow-ups and blow-downs using the interpretation of X as a moduli space of vector bundles on S. In view of the relation among Y , X and P 4 , understanding one of the involutions helps describe the behaviour of the others.
By the analogy of Y and S, one expects that the action of ι Y on Y has similar properties as the action of ι S on S, where the latter is well understood (see for example [Dol12,8.8.2]). To emphasize their analogy, we recall that the Bertini involution ι S on S can be described as follows.
The bianticanonical system |−2K S | is base-point-free and defines a 2:1-cover with image a quadric cone in P 3 . The Bertini involution ι S is then defined to be the associated covering involution. By construction, the Bertini involution ι S on S preserves every element of |−2K S |. Since a divisor D ∈ |−K S | defines an element 2D ∈ |−2K S |, we see that ι S preserves every divisor in |−K S |. In view of the abstract construction of ι Y on the Fano fourfold Y , the same methode cannot be applied to decide whether ι Y preserves every divisor in |−K Y |. However, by analysing the anticanonical map of Y , we show that the same property holds for Y . To understand the Bertini involution ι Y on the Fano fourfold, our approach is analysing its behaviour on a special surface W Y which is invariant by ι Y . This surface W Y is the strict transform of the cubic scroll swept out by the pencil of elliptic normal quintics in P 4 through the 8 blown up points; in particular, it contains the curve R Y . Inspired by the similarity with degree-one del Pezzo surfaces, we study the morphism defined by the restricted bianticanonical system of Y on W Y , and we give the following description of ι Y restricted to W Y .
Proposition 1.4. The Bertini involution ι Y of the Fano fourfold Y := M S,−K S preserves the surface W Y , and its restriction ι Y | W Y on W Y is the biregular involution defined by the double covering where V 2,4 ≃ F 2 is a rational normal scroll of bidegree (2, 4). In particular, the Bertini involution ι Y acts as the identity on the curve R Y and ι Y induces an involution on each elliptic fibre F Y of W Y → P 1 . Furthermore, there exists a smooth curve R ′ ∼ 3(R Y + F Y ) of genus 4 on the surface W Y , such that R ′ is disjoint from R Y and contained in the fixed locus of ι Y .
Since R Y is contained in the fixed locus of the Bertini involution ι Y , the involution can be lifted to the blow-upỸ of Y along the curve R Y . We establish the relation between the resolved anticanonical map and the lifted involution onỸ as follows.
be the morphism defined by the base-point-free linear system |µ * (−K Y ) − E|. Then f has generically degree 4 with image Q a smooth quadric hypersurface in P 5 , and f contracts the strict transform of the surface W Y to a conic in P 5 . Moreover, f | E : E → f (E) is a finite birational morphism such that the image f (E) has degree 4 in P 5 . Furthermore, the Bertini involution ι Y of Y can be lifted toỸ , and the lifted involution ιỸ acts as the identity on E. Moreover, f factors through the quotientỸ /ιỸ : As open questions, one may like to understand the quotientỸ /ιỸ , the geometric interpretation ofỸ /ιỸ → Q, and to describe completely the fixed locus of ιỸ (see Remark 4.15).
Plan. We briefly explain the organisation of the paper. In Section 2, we summarise some results in [CCF19], including the geometry of the Fano model Y := M S,−K S , the connection between the blow-up X of P 4 at 8 points and the degree-one del Pezzo surface S, and the relation between X and Y . We finish by recalling some basic properties of the Bertini involution of a degree-one del Pezzo surface.
In Section 3, we investigate the anticanonical system |−K Y | and the bianticanonical system |−2K Y |. We prove Proposition 1.1 by an additional analysis on the simplicial facets of the cone of effective divisors on Y . We also give some auxiliary results on |−K Y | and |−2K Y |, which serve as key ingredients in the study of the Bertini involution of Y .
In Section 4, we study the action of the Bertini involution of Y . Subsection 4.1 is devoted to the proof of Proposition 1.4. We study the morphism defined by the bianticanonical system |−2K Y | restricted to the surface W Y . Computations by Macaulay2 show that the image of W Y is a surface of degree 6 in P 7 , which helps us to describe completely the morphism; in particular, the morphism is finite of degree 2 and gives an involution on the surface W Y . By examining the action of this covering involution, we show that it coincides with the Bertini involution ι Y restricted to the surface W Y .
In Subsection 4.2, we study the geometry of the anticanonical map of Y . Computations by Macaulay2 show that the image of Y by the antincanonical map is a smooth quadric hypersurface Q in P 5 . We are then ready to prove Theorem 1.3. The strategy is to prove by contradiction: we suppose that ι Y does not preserve every divisor in |−K Y |. We show that in this case, ι Y induces a non-trivial involution ι Q on Q. We then obtain a contradiction by analysing the fixed locus of the induced involution ι Q and by studying the geometry of a special sub-linear systems of |−K Y | consisting of divisors containing the surface W Y . Theorem 1.5 is obtained as a consequence of the study to prove Theorem 1.3.
In Appendix A, we include the code for several computations in Section 3 and Section 4 using the software system Macaulay2.
Acknowledgements. This project was initiated during my stay in Turin. I heartily thank Cinzia Casagrande for her hospitality and fruitful conversations. I would like to express my sincere gratitude to my supervisor, Andreas Höring, for his patient guidance, his constant support and valuable suggestions. I also thank Daniele Faenzi for his help on computations, and Susanna Zimmermann for interesting discussions.
Many results in this paper are based on computations using Macaulay2. I would like to thank the developers for making their software open-source. I thank the IDEX UCA JEDI project (ANR-15-IDEX-01) and the MathIT project for providing financial support.

Preliminaries
We fix S a general del Pezzo surface of degree one. Let M S,L be the moduli space of semi-stable (with respect to L ∈ Pic(S) ample) rank-two torsion free sheaves F on S with c 1 (F) = −K S and c 2 (F) = 2. Then it follows from the classical properties of the determinant line bundle that for the polarisation L = −K S , the moduli space Y := M S,−K S is Fano.
For the degree-one del Pezzo surface S, we introduce the following notions (see [CCF19, Sect. 2.1]). A conic on S is a smooth rational curve such that −K S · C = 2 and C 2 = 0. Every such conic yields a conic bundle S → P 1 having C as fibre. There are 2160 conics (as classes of a curve) in H 2 (S, Z). A big divisor h on S which realises S as the blow-up σ : S → P 2 at 8 distinct points is called a cubic. We have h = σ * O P 2 (1). There are 17280 cubics (as classes of a curve) in H 2 (S, Z).
Notation 2.1. Given a cubic h, we use the following notation: • σ h : S → P 2 is the birational map defined by h • q 1 , . . . , q 8 ∈ P 2 are the points blown up by σ h • e i ⊂ S is the exceptional curve over q i , for i = 1 . . . , 8 • C i ⊂ S is the transform of a general line through q i , so that C i ∼ h − e i , for i = 1, . . . , 8 • ℓ ij ⊂ S is the transform of the line q i q j ⊂ P 2 , so that ℓ ij ∼ h − e i − e j and ℓ ij is a (−1)-curve, for 1 ≤ i < j ≤ 8.
yields an isomorphism between E and Eff(Y ), where E is the subcone of Nef(S) generated by the conics: Hence, the cone Eff(Y ) has 2160 extremal rays, each generated by a fixed divisor E C , where C ⊂ S is a conic. Moreover, given a cubic h, (2h + K S ) ⊥ ∩ E is a simplicial facet (i.e. a face of codimension one) of E, generated by the conics C i for i = 1, . . . , 8 (notations as in Notation 2.1). Hence, the fixed divisors E C i for i = 1, . . . , 8 generate a simplicial facet of Eff(Y ). . The cone of effective curves NE(Y ) has 240 extremal rays, and is isomorphic to NE(S). If ℓ is a (−1)-curve on S, the corresponding extremal ray of NE(Y ) is generated by the class of a line Γ ℓ in P ℓ ∼ = P 2 ⊂ Y . The corresponding elementary contraction is a small contraction, sending P ℓ to a point.
The determinant map ρ also relates the two automorphism groups Aut(Y ) and Aut(S). By [CCF19, Thm. 1.9], the map ψ : Aut(S) → Aut(Y ) given by ψ(φ)[F] = [(φ −1 ) * F], for φ ∈ Aut(S) and [F] ∈ Y , is a group isomorphism. In particular, Aut(Y ) is finite; if S is general, then We have a commutative diagram: (1) Finally, motivated by the analogy with del Pezzo surface of degree one, the study of the base loci of the anticanonical and the bianticanonical linear systems of Y gives the following: Theorem 2.4 ([CCF19], Thm. 1.10). The linear system |−K Y | has a base locus of positive dimension, while the linear system |−2K Y | is base point free.

2.2
The blow-up X of P 4 at 8 general points 2.2.1 Degree one del Pezzo surfaces and blow-ups of P 4 in 8 points For S = Bl q 1 ,...,q 8 P 2 and X = Bl p 1 ,...,p 8 P 4 the blow-ups respectively of P 2 and P 4 at 8 general points, there is a classical association between these two varieties due to Gale duality. The following is summarised from [CCF19, 2.21]; for further details of the association, we refer to [CCF19,2.18].
Let h be a cubic on S. We associate to (S, h) a blow-up X of P 4 in 8 points in general linear position as follows.
Conversely, let X be a blow-up of P 4 in 8 general points. Differently from the case of surfaces, the blow-up map X → P 4 is unique and thus X determines p 1 , . . . , p 8 ∈ P 4 up to projective equivalence. The 8 points p 1 , . . . , p 8 ∈ P 4 in turn determine q 1 , . . . , q 8 ∈ P 2 up to projective equivalence, and thus a pair (S, h) such that X ∼ = X (S,h) . The pair (S, h) is unique up to isomorphism, therefore S is determined up to isomorphism, and h is determined up to the action of the automorphism group Aut(S) on cubics.

From the blow-up X to the Fano model Y
We recall the explicit relation between X and Y : Lemma 2.5 ([CCF19], Lem. 5.18). The birational map ξ : X Y is the composition of 36 (Kpositive) flips: first the flips of L ij for 1 ≤ i < j ≤ 8, and then the flips of Γ k for k = 1, . . . , 8. There is a commutative diagram: where X → X is the blow-up of the curves L ij and Γ k , with every exceptional divisor isomorphic to P 1 × P 2 with normal bundle O(−1, −1), and X → Y is the blow-up of 36 pairwise disjoint smooth rational surfaces.
Notation 2.6. We use the following notation: We will sometimes write ξ h : X h Y to stress that X h and ξ h depend on the chosen cubic h (while Y does not). Denote by η h the composition map:

The Bertini involution of S
We recall some basic properties of the Bertini involution of a del Pezzo surface of degree one.
Proposition 2.7 ([Dol12],Thm. 8.3.2). Suppose that S is a del Pezzo surface of degree 1. Then (i) |−K S | is a pencil of genus 1 curves with smooth general member and one base point; (ii) |−2K S | is base-point-free and defines a morphism φ |−2K S | : S → P 3 which is finite of degree 2 with image Q a quadric cone.
The Bertini involution ι S : S → S is the biregular involution defined by the double covering For S general, ι S is the unique non-trivial automorphism of S. The pull-back ι * S acts on Pic(S) (and on H 2 (S, R)) by fixing K S and acting as −1 on K ⊥ S (see [Dol12, §8.8.2]). This yields: (2) The fixed locus of ι S is a smooth irreducible curve of genus 4 isomorphic to the branch curve of the double cover and the base point of |−K S |. The fixed curve belongs to the linear system | − 3K S |.

Anticanonical and bianticanonical linear systems of the Fano model Y
Let S be a degree-one del Pezzo surface, and Y := M S,−K S be the associated Fano fourfold. To analyse the anticanonical linear system |−K Y |, we introduce a special surface as follows.
Let W ′ ⊂ X be the transform of the cubic scroll W ⊂ P 4 . By [CCF19,(7.3)], we have the following diagram: where η : W ′ → W is the blow-up of p 1 , . . . , p 8 , so the composition α ′ := α • η : W ′ → P 2 is the blow-up of q 0 , . . . , q 8 . Thus W ′ is isomorphic to the blow-up of S in the base point of |−K S |. Hence, there is an elliptic fibration π : W ′ → P 1 , where the smooth fibres are the transforms of the elliptic normal quintics through p 1 , . . . , p 8 in P 4 , and every fibre is integral.
is a fibre of the elliptic fibration, and R ⊂ W ′ is a (−1)-curve and a section of the elliptic fibration. The curves R and F satisfy −K X · R = −K X · F = 1 and Moreover, let R 4 ⊂ P 4 be the images of R under η : W ′ ⊂ X → W ⊂ P 4 (see diagram (3)). Then R 4 is a smooth rational quintic curve through p 1 , . . . , p 8 Corollary 3.4 ([CCF19], Cor. 7.6). The base locus of |−K X | contains the smooth rational curve R, and the base locus of |−K Y | contains the smooth rational curve ξ(R).
We denote by R Y the smooth rational curve ξ(R) contained in the base locus of |−K Y |, and F Y a fibre of the elliptic fibration W Y → P 1 .
Lemma 3.7. The base locus of the anticanonical system |−K Y | is disjoint from the surfaces P ℓ ij and P e k , for 1 ≤ i < j ≤ 8 and k = 1, . . . , 8.
Proof. Consider the commutative diagram in Lemma 2.5: where p :X → X is the blow-up of X along the curves L ij and Γ k with every exceptional divisor isomorphic to P 1 ×P 2 , and q :X → Y is the blow-up of 36 pairwise disjoint smooth rational surfaces P ℓ ij and P e k , for 1 ≤ i < j ≤ 8 and k = 1, . . . , 8.
Suppose by contradiction that there exists a base point y of |−K Y | contained in some flipped surface that we denote by P (which is one of the surfaces P ℓ ij or P e k ). Denote by C ⊂ X the corresponding flipping curve (which is one of the curves L ij or Γ k ).
Let E be the sum of exceptional divisors over L ij for 1 ≤ i < j ≤ 8 and over Γ k for k = 1, ..., 8. Since Let E y ≃ P 1 be the exceptional fibre inX above y. Then E y is contained in Bs |q * (−K Y )| = Bs |p * (−K X ) − E| and E y is mapped surjectively onto C.
Since the blow-up of the 8 points X = Bl p 1 ,...,p 8 → P 4 is an isomorphism near a general point of C, the base scheme of |−K X | is generically reduced along C by Remark 3.6. Hence, the linear system |p * (−K X ) − E| is base-point-free above the generic point of C. This contradicts the fact that Bs |p * (−K X ) − E| contains a curve which is mapped surjectively onto C.
Remark 3.8. More generally, the proof of Lemma 3.7 shows the following. Let X, Y be smooth projective fourfolds. Let ξ : X Y be an anti-flip. In [Kaw89, Thm. 1.1], Kawamata showed that for smooth projective fourfolds, there exists only one type of flip and it is obtained by blowing up a P 2 with normal bundle O P 2 (−1) ⊕2 (the exceptional locus of the blowing up is P 2 × P 1 ) and blowing down this P 2 × P 1 to P 1 . Thus ξ (anti-)flips a smooth curve C ⊂ X to a smooth surface P ⊂ Y . If Bs |−K X | is reduced in the generic point of C, then |−K Y | is base-point-free on P .
Corollary 3.9. The curve R Y is the unique base curve in Bs |−K Y | of anticanonical degree 1. Therefore, R Y is independent of the choice of cubic h.
Proof. Let C ⊂ Bs |−K Y | be a base curve contained in some exceptional divisor ξ(E i ), for i = 1, . . . , 8. LetC be its strict transform in X. By Lemma 3.7, the curve C is disjoint from the indeterminacy locus of ξ −1 . Hence, one has −K Then for every simplicial facet E C 1 , . . . , E C 8 of Eff(Y ) (notation as in Notation 2.1 and Proposition 2.2), there exists a unique Since B is distinct from R Y , we deduce that B is contained in some fixed divisor E C i by Remark 3.6. By the construction of the composition of flips ξ (see Lemma 2.5), the intersection of two fixed divisors E C j and E C k (for k = j) is the union of the flipped surfaces P ℓ jk and P e l for l = j, k. Hence, by Lemma 3.7, the fixed divisor E i containing B is unique.
Let h be a cubic. Let C i be a conic such that C i ∼ h − e i for i = 1, . . . , 8 (notation as in Notation 2.1). Let E i := E C i , where we use the notation of Proposition 2.2. By the same proposition, E 1 , . . . , E 8 generate a simplicial facet of Eff(Y ). Suppose by contradiction that there exists another component B distinct from R Y of the base locus of |−K Y |. Then by Corollary 3.10, we may suppose that Then we can check that and 2h ′ +K S is orthogonal to the above 8 conics. Moreover, h ′ is nef and big, and the corresponding birational map σ h ′ : S → P 2 contracts the 8 pairwise disjoint (−1)-curves ℓ jk , ℓ ik , ℓ ij , e l for l = i, j, k. Hence, h ′ is a cubic. This proves the claim.
We will repeatedly use Corollary 3.10 in the following.
• Consider the simplicial facet generated by • Finally, consider the simplicial facet generated by Then by what precedes, we know that B is contained in none of these 8 fixed divisors, which contradicts Corollary 3.10.
there are no embedded points. Indeed, given a cubic h, consider the birational map η h : Y P 4 . By Remark 3.6, the base scheme of |−K Y | is the reduced curve R Y with possible embedded points which have support in the 8 points of intersection with the 8 exceptional divisors of η h . By varying h, we may consider another map η h ′ : Y P 4 with other 8 exceptional divisors, so that we get 8 different points of intersection on R Y . Such a cubic h ′ exists because otherwise, there is a base point y on R Y such that for every simplicial facet E C 1 , . . . , E C 8 of Eff(Y ) the point y is contained in a unique E C i , and thus we obtain a contradiction by replacing B with y in the above paragraph. Hence, there is no embedded base point on R Y .
Proof of Corollary 1.2. Since the base scheme Bs |−K Y | is the smooth curve R Y by Proposition 1.1, we can apply [MM86, Prop. 6.8] which implies that a general member in |−K Y | is smooth.
In the rest of this section, we collect some auxiliary results which will be used in Section 4.
Lemma 3.11. For a general point x ∈ R 4 (notation as in Lemma 3.3), there exists a unique divisor in M which has multiplicity 3 at x: it is the secant variety of the elliptic normal quintic through the nine points p 1 , . . . , p 8 and x.
By varying x on R 4 , one obtains a one-dimensional family Sec of divisors in M with schemetheoretic intersection Bs Sec defined by the ideal b(Sec). Then  Suppose by contradiction that W Y depends on h. Then there exist two distinct surfaces W Y,h and W Y,h ′ . Let Bs M Y,3 be the scheme-theoretic intersection of the family M Y,3 . Then by Lemma 3.11, one has the following set-theoretic inclusion: Since W Y,h ′ contains the curve R Y which is generically in the locus where ξ −1 h : Y X h is an isomorphism, we deduce that W Y,h ′ is not contracted by ξ −1 h . Since the surface ξ −1 h (W Y,h ′ ) contains the curve R, this surface cannot be contained in any exceptional locus E i , i = 1, ..., 8 of X h → P 4 , and thus it cannot be contracted; we denote by W h ′ its image in P 4 . Therefore, Bs Sec contains two distinct surfaces W and W h ′ , which contradicts Lemma 3.11.
Since by Lemma 3.2 the surface W Y is disjoint from the indeterminacy locus of the map ξ −1 h : Y X h , which is a union of some of the loci P ℓ (depending on h), and W Y is the same for all h, we deduce that W Y is disjoint from every one of the loci P ℓ .
..,p 8 ), and the surface W Y is disjoint from the indeterminacy locus of η h by Lemma 3.2 and W Y is not contained in the exceptional locus of η h , we deduce that is surjective.
10) ⊗ I 6 p 1 ,...,p 8 ) and by the same argument as above, we deduce that is surjective.

The Bertini involution of the Fano model Y
Let S be a degree-one del Pezzo surface, and Y := M S,−K S be the associated Fano fourfold. In this section, we study the action of the Bertini involution ι Y on the Fano fourfold Y , which is analogous to the action of the Bertini involution ι S on the surface S. We first notice that by the diagram (1) and the behaviour of ι S described in (2), the invariant part of H 2 (Y, R) by the action of ι Y is RK Y .

Action of the Bertini involution on the surface W Y
In this subsection, we further our study of the involution ι Y by looking at its action on the surface W Y (which is the strict transform of the cubic scroll swept out by the pencil of elliptic normal quintics in P 4 ). The aim of this subsection is to prove Proposition 1.4. We start by showing that the surface W Y is invariant by the Bertini involution ι Y .
Lemma 4.1. The Bertini involution ι Y preserves the curve R Y and the surface W Y . Moreover, Proof. Since ι Y preserves the family of divisors in the anticanonical system |−K Y |, the involution ι Y preserves the base locus of |−K Y |. Thus ι Y (R Y ) = R Y by Proposition 1.1. Let x be a general point in R Y . Then by Lemma 3.11, there exists a unique divisor in |−K Y | having multiplicity 3 at x: it is the strict transform in Y of the secant variety of the elliptic normal quintic through p 1 , . . . , p 8 and η h (x) in P 4 . In particular, this divisor has multiplicity 3 along the elliptic fibre of W Y through x. By varying x in R Y , this gives a one-dimensional family M Y,3 of divisors in |−K Y |, which is preserved by ι Y . On the other hand, the intersection of these divisors is the surface W Y , so W Y is preserved by ι Y . Let D 1 ∈ M Y,3 and D 2 = ι Y (D 1 ) ∈ M Y,3 . Let F 1 (resp. F 2 ) be the elliptic fibre of W Y along which D 1 (resp. D 2 ) has multiplicity 3. Then ι Y (F 1 ) = F 2 , and thus ι Y preserves the family of elliptic fibres of

Now we investigate the morphism defined by the linear system
is a rational normal scroll of bidegree (2, 4). There is a non-trivial involution i of W Y such that φ = φ•i. Moreover, i is the identity on R Y and i induces an involution on each elliptic fibre of W Y .
Proof. Since h 0 (W Y , O W Y (−2K Y )) = 8 (see Lemma 3.13), and |−2K Y | is base-point-free by Theorem 2.4, the linear system Claim. V is a surface of degree 6 in P 7 , the image of an elliptic fibre F Y by φ is a line and the image of R Y by φ is a conic.
Since the restriction morphism In P 4 , let 2M be the linear system of hypersurfaces of degree 10 with multiplicity at least 6 at the 8 general points p 1 , ..., p 8 . Consider the map φ 2M defined by the linear system 2M. Then by Macaulay2 (see Listing 8), the image of the surface W by φ 2M is a surface of degree 6, the image of an elliptic normal quintic through the 8 points by φ 2M is a line and the image of the rational quintic R 4 through the 8 points by φ 2M is a conic. This proves the claim.
, and the image of W Y by φ is of degree 6, we deduce that φ is of degree 2. As −K Y is ample, the morphism φ does not contract any curve and thus it is a finite morphism of degree 2.
Since the linear system |−2K Y | W Y | has no fixed divisor, the image V is not contained in any hyperplane of P 7 (see for example [Bea96, II.6]), i.e. V is non-degenerate. Hence, V is a nondegenerate irreducible surface of degree 6 (variety of minimal degree) in P 7 , and by [GH94,p. 525] we deduce that V is a rational normal scroll V k,l of bidegree (k, l), with 0 ≤ k ≤ l and k + l = 6. In particular, V is isomorphic to one of the following: a cone over a rational normal curve of degree 6, P 1 × P 1 , or a Hirzebruch surface F l−k , where the minimal section is mapped to the rational normal curve of degree k, and the fibres are mapped to lines. Therefore, φ is a finite morphism between two normal surfaces and by [Fuj83,(2

.3)], there is a non-trivial involution
Since the restriction of φ to a general fibre F Y induces a finite morphism from an elliptic curve to a line l ⊂ V , which cannot be an isomorphism, we deduce that φ −1 (l) = F Y as φ is of degree 2. Hence, i induces an involution on F Y . Since is torsionfree (this is because W Y is isomorphic to P 2 blown up at 9 points), we deduce that i We claim that R Y is contained in the ramification locus of φ. Indeed, suppose that R Y is not contained in the ramification locus of φ. Then there exists a curve C ⊂ V such that R Y = φ * (C). As R Y is a (−1)-curve on W Y , one has i.e. C 2 = − 1 2 . Hence, C is not Cartier on V , i.e. V is singular. In view of the classification of minimal degree varieties, we see that V is a cone. But there is no curve with negative selfintersection number on a cone, which leads to a contradiction. Therefore, R Y is in the ramification locus. As φ is a double cover, we deduce that i is the identity on R Y .
Let C = φ(R Y ). Since R Y is contained in the ramification locus of φ, and every point in R Y has ramification index 2, one has Since R Y is a (−1)-curve on W Y , one has C 2 = −2. Therefore, V = V 2,4 ≃ F 2 , and φ(R Y ) is minimal section of F 2 which is a conic.
Remark 4.3. Since φ is a finite morphism of degree 2 between smooth surfaces, the ramification locus is a smooth divisor on W Y (see [Fuj83, (2.5)]). Let e be the minimal section of V ≃ F 2 and f be a fibre of V . Let D be the ramification divisor. Then Let B ⊂ V be the branch locus. Then D = 1 2 φ * B and thus B ∼ 4e + 6f . As e is contained in the branch locus, we can write B = e + B 1 , where B 1 is a smooth curve disjoint from e. Then B 1 ∼ 3e + 6f . Notice that B 1 is irreducible. Indeed, suppose that B 1 has at least two disjoint irreducible components. Then we can decompose B 1 as with 0 ≤ b ≤ 6 and (e + bf ) · (2e + (6 − b)f ) = 0. Hence b = −2, which leads to a contradiction. Hence Finally, we compare the action of the two automorphisms i and ι Y | W Y on W Y .
Proof. For i = 1, . . . , 8, by Macaulay2 (see Listing 9), there exists a unique hypersurface of degree 10 with multiplicity at least 7 at the point p i and multiplicity at least 6 at p j for j = i. Moreover, this hypersurface does not contain the surface W . Therefore, the linear system Since W ′ is disjoint from the indeterminacy locus of ξ h , it is equivalent to show that R, F and e i for i = 1, . . . , 8 form a basis of H 2 (W ′ , R). We have the following diagram (see (3)): where α is the blow-up of P 2 at one point and η is the blow-up of W at p 1 , . . . , p 8 . Moreover, let e 0 ⊂ W be the (−1)-curve and f 0 ⊂ W be a fibre of the P 1 -bundle on W , then by Lemma 3.1 and Lemma 3.3, one has F ∼ η * (2e 0 + 3f 0 ) − 8 i=1 e i and R ∼ η * (e 0 + 4f 0 ) − 8 i=1 e i . Therefore, R, F and e i for i = 1, . . . , 8 form a basis of H 2 (W ′ , R).
We have a group homomorphism Since R Y is a (−1)-curve on W Y , by blowing down R Y , we obtain a rational surface S ′ with (−K S ′ ) 2 = 1, and the curve R Y is contracted to a point x 0 ∈ S ′ . We denote by β : W Y → S ′ the blow-up of S ′ at x 0 . Since −K W Y is nef, we obtain that −K S ′ is nef by the projection formula (see for example [Har77, Appendix A, A4]). Moreover, since every fibre of W Y → P 1 is integral, there is no K S ′ -trivial curve. Hence, S ′ is a del Pezzo surface of degree one. By [Dol12,Prop. 8.2.39], the homomorphism ρ 2 : Aut(S ′ ) → Aut H 2 (S ′ , R) is injective.
Let Aut(x 0 , S ′ ) be the subgroup of automorphisms in Aut(S ′ ) fixing the point x 0 . Then R)) such that G 1 ≃ Aut(H 2 (S ′ , R)). Hence, we have the following diagram: Since ρ 2 is injective, the restriction ρ 1 : Proof of Proposition 1.4. The first paragraph follows from Lemma 4.1, Proposition 4.2 and Proposition 4.5. The second paragraph follows from Remark 4.3.

Action of the Bertini involution on the anticanonical system
In this subsection, we study the action of the involution ι Y on the anticanonical system |−K Y |. This is closely related to the anticanonical map of Y = M S,−K S .
Lemma 4.6. Let µ :Ỹ → Y be the blow-up of Y along the curve R Y which is the base scheme of |−K Y |. Let E be the exceptional divisor andD be the strict transform of a general member with image Q a smooth quadric hypersurface, and f has generically degree 4. We have the following commutative diagram: The following statements hold: (i) The Bertini involution ι Y can be lifted to an involution ιỸ ofỸ , which preserves the exceptional divisor E and induces an involution on each P 2 above a point of R Y .
(ii) The Bertini involution ι Y induces an involution ι P 5 of P 5 , which preserves the quadric hypersurface Q. Denote by ι Q its restriction on Q. Then Proof. In P 4 , let M be the linear system of quintic hypersurfaces with multiplicity at least 3 at 8 general points. Then by Macaulay2, the image of P 4 by the map defined by M is a smooth quadric hypersurface Q in P 5 . Let E be the exceptional divisor of f . Since one has D 4 = 8. Hence φ |−K Y | (and also f ) has generically degree 4. (i) Follows from the fact that R Y is contained in the fixed locus of ι Y (see Proposition 1.4).
Recall that we have a special surface W Y ⊂ Y containing R Y , which is an elliptic fibration W Y → P 1 with fibre F Y . With the same notation as in Lemma 4.6, we describe the image of W Y in Q ⊂ P 5 .
Lemma 4.8. Every elliptic fibre F Y (resp. its strict transformF Y ⊂Ỹ ) is contracted by φ |−K Y | (resp. by f ). Moreover, the image of the surface W Y (resp. its strict transformW Y ⊂Ỹ ) is a conic C in Q ⊂ P 5 . Furthermore, the curveR Y :=W Y ∩ E is contained in the fixed locus of ιỸ , and the conic C is contained in the fixed locus of ι Q .
Proof. Since −K Y · F Y = 1, one hasD ·F Y = 0, whereD is the strict transform of a general member D ∈ |−K Y |. Hence f contracts the elliptic fibres ofW Y . As Therefore, the morphism f sendsW Y to a conic in P 5 .
By Lemma 4.6 (i), ιỸ induces an involution on each fibre P 2 of µ| E : E → R Y . Since W Y is preserved by ι Y by Proposition 1.4, its transformW Y ⊂Ỹ is preserved by ιỸ . Therefore, the curveR Y :=W Y ∩ E (which is a section of µ| E ) is invariant. Since R Y is in the fixed locus of ι Y by Proposition 1.4, it follows thatR Y is contained in the fixed locus of ιỸ . By Lemma 4.6 (ii), The rest of this subsection is devoted to the proofs of Theorems 1.3 and 1.5. To show that ι Y preserves every divisor in |−K Y |, our strategy is to exclude the other remaining case by analysing the anticanonical map.
where V 1 is the sub-vector space of global sections vanishing on the surface W Y , and V 2 is uniquely determined as eigenspace corresponding to the eigenvalue 1 or −1 of ι * Y , with dim V 1 = dim V 2 = 3. More precisely, ι * Y acts as Id or − Id on V i for i = 1, 2. Proof. In P 4 , let M be the linear system of quintic hypersurfaces with multiplicity at least 3 at 8 general points. Let M W be the sub-linear system of effective divisors in M containing the surface W . By Macaulay2 (see Listing 5), one has b(M W ) = b(Sec) (see Lemma 3.11 for notation) for the base ideals. Moreover, by Macaulay2 (see Listing 6), a random divisor in M W is singular along two elliptic normal quintic curves E p , E q through the 8 blown up points (the two elliptic curves may coincide, in which case the divisor has multiplicity at least 3 along this elliptic curve, and in fact the divisor is the secant variety of the elliptic curve). Moreover, there exists a unique divisor in M W which is singular along E p and E q , as H 0 (P 4 , ) be the sub-vector space of global sections vanishing on the surface W Y . Let |V 1 | be the corresponding sub-linear system (i.e. the linear system of effective divisors in |−K Y | containing the surface W Y ). Then ι Y preserves the family of divisors in |V 1 |, as ι Y preserves the surface W Y by Proposition 1.4. Since W Y is disjoint from the indeterminacy locus of the map η h : Y P 4 , and the intersection of W Y with the exceptional locus of η h is the union of 8 points, we deduce that a general member in |V 1 | is singular along two elliptic fibres F Y,1 , F Y,2 of W Y , and there exists a unique divisor in |V 1 | which is singular along F Y,1 and F Y,2 . Since ι Y preserves every elliptic fibre F Y of W Y (see Proposition 1.4), we deduce that ι Y preserves every divisor in |V 1 |, i.e. the action of ι * Y on V 1 is Id or − Id. By Lemma 3.13(i), we have the following short exact sequence: , we deduce that V 2 can be uniquely determined as the eigenspace corresponding to the eigenvalue 1 or −1 of ι * Y .
) be the sub-vector space of global sections which are the linear spans ofs ij with j = 1, 2, 3, wheres ij is the strict transform of the global section s ij ∈ V i . Hence, if a point y ∈Ỹ is fixed by ιỸ , then by repeating the argument in above paragraph, we obtain y ∈ Bs | V 1 | or y ∈ Bs | V 2 |. To summarise, we have the following corollary.
Corollary 4.10. Suppose that ι Q is not the identity. If a point y ∈Ỹ is fixed by ιỸ , then y ∈ Bs | V 1 | or y ∈ Bs | V 2 |.

Recall that we have the normal bundle
Let l be an exceptional curve of µ and γ be the curve which generates the other extremal ray Γ of NE(E) such that −K E · γ is the length of Γ. Then Moreover,R Y ∼ l + γ.
With the same notation as in Lemma 4.6, we may describe the image f (E) as follows.
Remark 4.11. The exceptional divisor E is isomorphic to the blow-up B of P 3 at a line (and B is embedded in P 1 × P 3 with bidegree (1, 1)). LetD ⊂Ỹ be the strict transform of a general member D ∈ |−K Y |. Then (D| E ) 3 = 4, hencẽ D| E ∼ ξ +F E is very ample, with h 0 (E, O E (D)) = 7. Thus the corresponding linear system embeds B in P 6 as a hyperplane section of the Segre embedding of P 1 × P 3 in P 7 , and B has degree 4. Hence, f | E is given by the projection of B from a point x outside B in P 6 (in fact, it is given by a sub-linear system of |D| of dimension 5, which is still base-point-free).
If the point x is general, then the projection is birational and the image has degree 4 in P 5 . There could be special point x such that the projection has degree 2, and the image is a 3-dimensional quadric in P 5 . In any case, the image of a fibre F E is a plane in P 5 .
Lemma 4.12. Suppose that ι Q is not the identity. Then ιỸ | E is not the identity, and the following statements hold: (i) The fixed locus of ιỸ | E is the disjoint union S E ∪ C 2 , where S E = Bs | V 1 | ∩ E is the unique member in |ξ −F E | isomorphic to P 1 ×P 1 , and C 2 = Bs | V 2 |∩E is a curve satisfying C 2 ∼ l+γ which is mapped surjectively to R Y .
(ii) The fixed locus of ι P 5 is two disjoint planes P 2 1 ∪ P 2 2 such that f (S E ) = P 2 1 and that f (C 2 ) is a conic contained in P 2 2 . Furthermore, f (E) is a 3-dimensional quadric in P 5 .
Proof. Suppose by contradiction that ιỸ | E is the identity. Then by Corollary 4.10, one has E ⊂ , 2, this contradicts to the fact that there is no divisor in |−K Y | having multiplicity at least 2 along R Y by Macaulay2 (see Listing 11).
(i) Since ιỸ | E is not the identity and ι Y | R Y is the identity, we have that ιỸ | F E is not the identity. Thus the fixed locus of ιỸ | F E is the disjoint union of a point and a line.
We first describe the base locus of | V 1 |. Since by Macaulay2 (see Listing 5), one has b(M W ) = b(Sec) (see Lemma 3.11 for notation) for the base ideals. Thus the base locus of |V 1 | contains the surface W Y with multiplicity 2 by Lemma 3.11. Therefore, the base locus of | V 1 | contains the strict transformW Y ⊂Ỹ . Moreover, since a general member in |V 1 | is singular along two elliptic fibres of W Y , a local computation shows that every member in | V 1 | contains two fibres F E above the two points on R Y where it is singular. AsD| E ∼ ξ + F E , we deduce that the unique member Now we describe the base locus of | V 2 |. Since |µ * (−K Y ) − E| is base-point-free, Bs | V 2 | is disjoint from the surface S E . Let D 2 be a general member in |V 2 |. Since D 2 does not contain the surface W Y , and D 2 | W Y = R Y + 2F Y , we deduce that the intersection of the singular locus Sing D 2 with the curve R Y contains at most one point (which is a singularity of multiplicity two). Hence a general member in | V 2 | contains at most one fibre Claim. Bs | V 2 | ∩ E has dimension at most one. Suppose that there is a surface S 2 ⊂ Bs | V 2 | ∩ E. SinceD| E ∼ ξ + F E , one has S 2 ∈ |ξ| or S 2 ∈ |ξ + F E |. As ξ ·R Y = F E ·R Y = 1, one has S 2 ·R Y > 0, which contradicts the fact that S 2 is disjoint from S E . This proves the claim.
Note that Bs | V 2 | ∩ E has dimension one. This is because the fixed locus of ιỸ | F E is the disjoint union of a point and a line, and Bs | V 1 | ∩ F E is a line. Thus by Corollary 4.10, the fixed point disjoint from the fixed line is contained in Bs | V 2 | ∩ F E . Therefore, Bs | V 2 | ∩ E is a curve which is mapped surjectively to R Y .
(ii) Since ι Q is not the identity (i.e. ι P 5 is not the identity), the fixed locus of ι P 5 is the union of two disjoint sub-linear spaces P i ∪ P j with i + j = 4. Therefore, the fixed locus of ι P 5 is two disjoint planes P 2 1 and P 2 2 by the equation (5). By Remark 4.11, f (S E ) has dimension 2. Hence, f (S E ) ⊂ Q is one of the two planes P 2 1 and P 2 2 contained in the fixed locus of ι P 5 . We may denote f (S E ) = P 2 1 . Following the discussion in Remark 4.11, we now describe the map f | S E : the restricted linear system |D| E | S E embeds the surface S E in P 3 as the Segre embedding of P 1 × P 1 in P 3 , and the image S ′ E has degree 2. Hence, f | S E is given by the projection of S ′ E from a point outside S ′ E in P 3 . The projection has degree 2 and the image is the plane P 2 1 . Note that in Remark 4.11, the projection of B from a point x outside B in P 6 cannot be birational, as the projection of S ′ E ⊂ B from the point x in P 3 ⊂ P 6 has degree 2. Therefore, f (E) is a 3-dimensional quadric in P 5 .
Corollary 4.13. The involution ι Q is the identity, and thus the Bertini involution ι Y preserves every divisor in |−K Y | and f factors through the quotientỸ /ιỸ via the lifted involution ιỸ .
Proof. Suppose by contradiction that ι Q is not the identity. We use the notation as in Lemma 4.12.
Since the restricted linear system |D| E | S E embeds the surface S E in P 3 as the Segre embedding of P 1 × P 1 in P 3 with image S ′ E a quadric surface, the map f | S E is given by the projection of S ′ E from a point outside S ′ E in P 3 . The projection has degree 2, and f (S E ) = P 2 1 by Lemma 4.12. Therefore, f | S E : S E ≃ P 1 × P 1 → P 2 1 is a double cover branched over a non-singular conic ∆ in P 2 1 ; moreover, the image of any line on S E is a tangent line to ∆, and conversely the preimage of each tangent line on ∆ is two lines on S E , one from each ruling.
Let D ∈ |V 1 | be a general member andD ∈ | V 1 | be its strict transform. Then by Lemma 4.12,D ∩ E contains the surface S E and two distinct P 2 (denoted by F E 1 and F E 2 ) above the two points on R Y where D is singular. Thus f (D) contains f (F E 1 ) =: Π 1 and f (F E 2 ) =: Π 2 which are two planes in P 5 by Remark 4.11. Moreover, Π 1 and Π 2 are distinct. This is because F E 1 ∩ S E and F E 2 ∩ S E are two distinct lines of a same ruling of S E ≃ P 1 × P 1 , and thus their images in P 2 1 = f (S E ) are two distinct tangent lines to ∆ by the above discussion. Therefore, f (D) ∩ f (E) contains three distinct planes P 2 1 , Π 1 , Π 2 . This contradicts to the fact that f (D) is a hyperplane in P 5 and f (E) is a 3-dimensional quadric in P 5 (so that their intersection is a surface of degree 2 in P 5 ).
Therefore, ι Q is the identity. By Lemma 4.6 (ii), one has f = f • ιỸ . Thus f factors through the quotientỸ /ιỸ . Corollary 4.14. The morphism f | E : E → f (E) is birational, and f (E) has degree 4 in P 5 . Moreover, the restricted involution ιỸ | E is the identity.
Proof. By Remark 4.11, either f | E has degree 2 and the image is a 3-dimensional quadric in P 5 , or f | E is finite birational and the image has degree 4 in P 5 . We will show that the first case cannot happen.
Suppose that f | E has degree 2 and f (E) is a 3-dimensional quadric in P 5 . We will follow the same argument as in the proof of Corollary 4.13. By Lemma 4.12 (with the same notation), for a general member D ∈ |V 1 |, its strict transformD ∈ | V 1 | contains the surface S E ⊂ E. Moreover,D contains the two distinct fibres (denoted by F E 1 and F E 2 ) of µ| E : E → R Y above the two points on R Y where D is singular. Hence, f (D) ∩ f (E) contains the surface f (S E ) and the two planes f (F E 1 ), f (F E 2 ) (which may coincide).
(a) If f | S E has degree 2, then the same argument as in the proof of Corollary 4.13 shows that (b) If f | S E has degree 1, then f (S E ) is either a non-normal surface or isomorphic to S E ≃ P 1 ×P 1 . Thus f (S E ) has degree at least 2 in P 5 .
This contradicts to the fact that f (D) ∩ f (E) is a surface of degree 2 in P 5 . Therefore, f | E is finite birational and f (E) has degree 4 in P 5 . Now suppose that ιỸ | E is not the identity. Since f | E = f | E • ιỸ | E by Corollary 4.13, we deduce that f | E has degree 2, which leads to a contradiction.
Proof of Theorem 1.3. Follows from Corollary 4.13.
Remark 4.15. The fixed locus of ιỸ is E ∪ Res, where Res has dimension at most 2 and its intersection with everyP ℓ is non-empty and zero-dimensional, whereP ℓ ⊂Ỹ is the strict transform of P ℓ (see notation in Proposition 2.3).
Proof. Let P ℓ ≃ P 2 be the exceptional locus of a small extremal contraction of Y . Then ι Y (P ℓ ) = P ι * S (ℓ) is also the exceptional locus of some small extremal contraction of Y and P ℓ intersects ι Y (P ℓ ) transversally at 3 points by [CCF19, Rem. 2.15 (c), Lem. 6.4]. Therefore, the intersection of P ℓ with the fixed locus of ι Y is non-empty and zero-dimensional.
As R Y ⊂ W Y is disjoint from P ℓ by Lemma 3.12, we deduce that E is disjoint fromP ℓ . Therefore, Res ∩P ℓ is non-empty and zero-dimensional. As every non-zero effective divisor in Y must have positive intersection with some extremal ray of NE(Y ), we deduce that Res has dimension at most 2. We compute the image of the elliptic normal quintic E1 via the map defined by the linear system of hypersurfaces of degree 10 through the 8 points with multiplicity at least 6: S2 = k [ u 0 . . u 28 ] ; ImE1 = ker map(R/E1 , S2 ,GG) ; dim ImE1 , degree ImE1 We compute the image of the rational quintic curve I5 via the map defined by the linear system of hypersurfaces of degree 10 through the 8 points with multiplicity at least 6: ImI5 = ker map(R/ I5 , S2 ,GG) ; dim ImI5 , degree ImI5 We compute the image of the surface W via the map defined by the linear system of hypersurfaces of degree 10 through the 8 points with multiplicity at least 6: ImW = ker map(R/W, S2 ,GG) ; dim ImW, degree ImW Listing 8: Some images by the bianticanonical map We compute the hypersurfaces of degree 10 with multiplicity at least 7 at the point I 0 and multiplicity at least 6 at the other 7 points: J I 0 0 = i n t e r s e c t ( I 0ˆ7 , I I 1 ) ; HI00 = saturate J I 0 0 ; GI00 = gens HI00 ; bet t i GI00 GG0 = submatrix ( GI00 , { 0 } ) ; IGG0 = i d e a l (GG0 ) ; And we obtain a unique such hypersurface of degree 10; now we check if this hypersurface contains the surface W: JW0 = i n t e r s e c t (W, IGG0 ) ; HW0 = saturate JW0; GW0 = gens HW0; bet t i GW0 Listing 9: Special member in the bianticanonical system We compute the image of P 4 via the map defined by the linear system of quintic hypersurfaces through the 8 points with multiplicity at least 3. JJ = minors ( 2 ,random(Rˆ{ 4 : 0 } ,Rˆ{ −2 , −3})); degree JJ genus JJ bet t i res JJ == bet t i res i d e a l (G1) S = k [ y 0 . . y 5 ] ; g = map(R, S , gens JJ ) ; K = ker g ; dim K degree K singularLocus variety K Listing 10: Image by the anticanonical map We check that there is no quintic hypersurfaces through the 8 points with multipilicity at least 3 and having multiplicity at least 2 along the smooth rational quintic curve I5: JRR = i n t e r s e c t ( Jˆ3 , I 5ˆ2 ) ; HRR = saturate JRR; GRR = gens HRR; bet t i GRR Listing 11: Quintics having multiplicity 2 along the smooth rational quintic base curve