A note on Seshadri constants of line bundles on hyperelliptic surfaces

We study the Seshadri constants of ample line bundles on hyperelliptic surfaces. We obtain new lower bounds and compute the exact values of the Seshadri constants in some cases. Our approach uses results of Serrano (Math. Z. 203:527–533, 1990), Harbourne and Roé (J. Pure Appl. Alg. 212:616–627, 2008), Bastianelli (Manuscripta Math. 130:113–120, 2009), Knutsen, Syzdek and Szemberg (Math. Res. Lett. 16:711–719, 2009).

1. Introduction. Seshadri constants measure how positive a line bundle is. They were introduced in 1992 by Demailly [9] 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., [4]. There exists an upper bound for the 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 . Regarding lower bounds, there are examples due to Miranda and Viehweg which show that the Seshadri constants of an ample line bundle can attain arbitrarily small positive values.
Let us now recall some results concerning Seshadri constants on surfaces with Kodaira dimension zero. In the appendix to [3], Bauer and Szemberg give an upper bound for the global Seshadri constant of an ample line bundle on an abelian surface and as a corollary they obtain that the Seshadri constant of such a line bundle is always rational. In [2] Bauer computes the Seshadri constants on all K3 surfaces of degree 4. This result is extended by Galati and 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. In Proposition 3.3 we point out a hyperelliptic surface type and a point at which the 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 an arbitrary ample line bundle on a hyperelliptic surface of type 1, and in Theorem 3.5 we provide a lower bound for the global Seshadri 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 bundle at r very general points on hyperelliptic surfaces.

Notation and auxiliary results.
Let us set up the notation and basic definitions. Our surfaces are always smooth irreducible projective varieties of dimension 2 defined over the field of complex numbers C, curves are irreducible subvarieties of dimension 1. By D 1 ≡ D 2 we denote the numerical equivalence of divisors D 1 and D 2 . We use the notation as in [16].

Seshadri constants.
Let X be a smooth projective variety and L a nef line bundle on X. We recall the definition of the Seshadri constant.
where the infimum is taken over all curves C ⊂ X passing through x. Let x 1 , . . ., x r be pairwise distinct points. The notion of the Seshadri constant of a line bundle at a point may be generalised to r points in the following way: where the infimum is taken over all 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 in very general position is equal to sup{ε(L, x 1 , . . . , x r )} where the supremum is taken over all choices of r distinct points x 1 , . . . , x r ∈ X (see [16,Example 5.1.11]). We denote the Seshadri constant of L at r points in very general position by ε(L, r). Let For background on Seshadri constants, we refer to an interesting overview [5].

Hyperelliptic surfaces.
Let us start with recalling the definition of a hyperelliptic surface.
where Φ and Ψ are the natural projections. Hyperelliptic surfaces were classified at the beginning of the 20th century by Bagnera and de Franchis [8], and independently by Enriques and Severi [10]. They showed that there are seven non-isomorphic types of hyperelliptic surfaces. These types are characterised by the action of G on B ∼ = C/(Zω ⊕ Z) (for details see, e.g., [7,VI.20]). The canonical divisor K S of each hyperelliptic surface is numerically trivial.
In 1990 Serrano [17] characterised the group Num(S) for each type of surface: (Serrano). A basis of the group of classes of numerically equivalent divisors Num(S) for each of type of surface and the multiplicities of the singular fibres in each case are the following: Let μ = lcm{m 1 , . . . , m s } and let γ = |G|. Notice that a basis of Num(S) consists of divisors A/μ and (μ/γ) B. Definition 2.6. We say that L is a line bundle of type (a, b) on a hyperelliptic surface, In Num(S) we have A 2 = 0, B 2 = 0, AB = γ. Due to [1, Proposition 5.2], we have a criterion for effectiveness of a divisor of type (0, b), i.e. a divisor numerically equivalent to b · (μ/γ) B, namely The following proposition holds: Note that: Remark 2.10. Each 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 the elliptic curve A/G. This contradicts the Riemann-Hurwitz formula.
For families of curves, we have a Xu-type lemma. The original version of this lemma was proved by Xu [19]. We will use the generalisation of Xu's Lemma obtained by Knutsen, Syzdek, Szemberg [15], and independently by Bastianelli [6]. 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 blowup at x 2 , . . ., x r , we have the following multi-point version of the Xu-type lemma.
Lemma 2.12. For a general curve C of the family C as above, we have By Remark 2.10 there are no rational curves on a hyperelliptic surface S hence for every curve C ⊂ S on we have gon( C) ≥ 2.

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).  Proof. Let C ≡ (α, β) denote a curve passing through a point x ∈ S with multiplicity m, m ≥ 1. We estimate the value of LC m from below. Depending on the position of the point x and on the type of the hyperelliptic surface, we have the following possibilities for C to be a 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 (for admissible values of k see Theorem 2.5 and Lemma 2.7). Then (2) C ≡ nA/μ ≡ (n, 0) and the point x lies on a fibre nA/μ, where n ∈ {1, 2} for a hyperelliptic surface of type 1 and 2; n ∈ {1, 2, 4} for type 3 and 4; n ∈ {1, 3} for type 5 and 6; n ∈ {1, 2, 3, 6} for type 7 (for admissible values of n see Theorem 2.5). Then (3) C ≡ (α, β), where α > 0 and β > 0, and x is an arbitrary point. Then by Bézout's theorem, intersecting C with a fibre B and with an appropriate fibre nA/μ depending on the position of the point x, we get: in case of a hyperelliptic surface of type 1, 3, 5, 7;  From the proof of Theorem 3.1, we immediately obtain the following corollary: On the other hand, it is not true that on each hyperelliptic surface the equality ε(L, x) = 1 holds for every x ∈ S.
Using the same method as presented in Proposition 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 into a large number of cases, and therefore we have decided to not 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. Hence if x lies on a singular fibre A/2, then ε(L, x) = min{a, b, a + b} = min{a, b}, and if x does not lie on any singular fibre A/2, then ε(L, x) = min{a, 2b, a 2 +b} = min{a, 2b}. Since L·(A/2) > 0, the assertion is proved. 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, i.e. 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

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., [5, Proposition 2.1.1]) that for smooth projective surfaces ε(L, r) ≤ L 2 r . The 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 surface. Let L be an ample line bundle on S. Then ε(L, r) ≥ L 2 r 1 − 1 8r , r ≥ 2.
Proof. The claim follows immediately from the Harbourne-Roé theorem (Theorem 2.3) with μ = 8. The point is to check that the assumptions of the theorem are satisfied with this particular constant. Turning into details, we need to check the following two conditions: (1) for every integer 1 ≤ m < 8, α 0 (L, m [r] ) ≥ m L 2 r − 1