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Given a rational elliptic surface X over an algebraically closed field, we investigate whether a given natural number k can be the intersection number of two sections of X. If not, we say that k is a gap number. We try to answer when gap numbers exist, how they are distributed and how to identify them. Our main tool is the Mordell–Weil lattice, which connects the investigation to the classical problem of representing integers by positive-definite quadratic forms. [ABSTRACT FROM AUTHOR]

We classify all Jacobian elliptic fibrations on K3 surfaces with finite automorphism group. We also classify all Jacobian elliptic fibrations with finite Mordell–Weil group on K3 surfaces with infinite automorphism group and 2-elementary Néron–Severi lattice. As part of the classification, we compute the lattice theoretic multiplicities of all Jacobian elliptic fibrations in both cases. [ABSTRACT FROM AUTHOR]

We show that the moduli space of U ⊕ ⟨ - 2 k ⟩ -polarized K3 surfaces is unirational for k ≤ 50 and k ∉ { 11 , 35 , 42 , 48 } , and for other several values of k up to k = 97 . Our proof is based on a systematic study of the projective models of elliptic K3 surfaces in P n for 3 ≤ n ≤ 5 containing either the union of two smooth rational curves or the union of a smooth rational curve and an elliptic curve intersecting at one point. [ABSTRACT FROM AUTHOR]

In this article, we study quasimaps to moduli spaces of sheaves on a $K3$ surface S. We construct a surjective cosection of the obstruction theory of moduli spaces of $\epsilon $ -stable quasimaps. We then establish reduced wall-crossing formulas which relate the reduced Gromov–Witten theory of moduli spaces of sheaves on S and the reduced Donaldson–Thomas theory of $S\times C$ , where C is a nodal curve. As applications, we prove the Hilbert-schemes part of the Igusa cusp form conjecture; higher-rank/rank-one Donaldson–Thomas correspondence with relative insertions on $S\times C$ , if $g(C)\leq 1$ ; Donaldson–Thomas/Pandharipande–Thomas correspondence with relative insertions on $S\times \mathbb {P}^1$ .

We prove a decomposition theorem for the nef cone of smooth fiber products over curves, subject to the necessary condition that their Néron–Severi space decomposes. We apply it to describe the nef cone of so-called Schoen varieties, which are the higher-dimensional analogues of the Calabi–Yau threefolds constructed by Schoen. Schoen varieties give rise to Calabi–Yau pairs, and in each dimension at least three, there exist Schoen varieties with nonpolyhedral nef cone. We prove the Kawamata–Morrison–Totaro cone conjecture for the nef cones of Schoen varieties, which generalizes the work by Grassi and Morrison.

We study the problem of classifying pencils of plane sextics up to projective equivalence via geometric invariant theory (GIT). In particular, we provide a complete description of the GIT stability of certain pencils of plane sextics called Halphen pencils of index two-classical geometric objects which were first introduced by G. Halphen in 1882. Inspired by the work of Miranda on pencils of plane cubics, we obtain explicit stability criteria in terms of the types of singular fibers appearing in their associated rational elliptic surfaces. [ABSTRACT FROM AUTHOR]

For the study of the Mordell–Weil group of an elliptic curve E over a complex function field of a projective curve B, the first author introduced the use of differential-geometric methods arising from Kähler metrics on H × C invariant under the action of the semi-direct product SL (2 , R) ⋉ R 2 . To a properly chosen geometric model π : E → B of E as an elliptic surface and a non-torsion holomorphic section σ : B → E there is an associated "verticality" η σ of σ related to the locally defined Betti map. The first-order linear differential equation satisfied by η σ , expressed in terms of invariant metrics, is made use of to count the zeros of η σ , in the case when the regular locus B 0 ⊂ B of π : E → B admits a classifying map f 0 into a modular curve for elliptic curves with level-k structure, k ≥ 3 , explicitly and linearly in terms of the degree of the ramification divisor R f 0 of the classifying map, and the degree of the log-canonical line bundle of B 0 in B. Our method highlights deg (R f 0) in the estimates, and recovers the effective estimate obtained by a different method of Ulmer–Urzúa on the multiplicities of the Betti map associated to a non-torsion section, noting that the finiteness of zeros of η σ was due to Corvaja–Demeio–Masser–Zannier. The role of R f 0 is natural in the subject given that in the case of an elliptic modular surface there is no non-torsion section by a theorem of Shioda, for which a differential-geometric proof had been given by the first author. Our approach sheds light on the study of non-torsion sections of certain abelian schemes. [ABSTRACT FROM AUTHOR]

We describe a method to compute mod ℓ Galois representations contained in the H e ´ t 2 of surfaces. We apply this method to the case of a representation with values in GL 3 (F 9) attached to an eigenform over a congruence subgroup of SL 3 . We obtain, in particular, a polynomial with Galois group isomorphic to the simple group PSU 3 (F 9) and ramified at 2 and 3 only. [ABSTRACT FROM AUTHOR]

We study torsion-free, rank 2 Higgs sheaves on genus one fibered surfaces, (semi)stable with respect to suitable polarizations in the sense of Friedman and O'Grady. We prove that slope-semistability of a Higgs sheaf on the surface implies semistability on the generic fiber. In the case of Higgs sheaves of odd fiber degree on elliptic surfaces in characteristic ≠ 2 , we prove that any moduli space of Higgs sheaves with fixed numerical invariants splits canonically as the product of the moduli space of ordinary sheaves (with the same invariants), and the space of global regular 1-forms on the surface. For elliptic surfaces with section in characteristic zero, and in the case arbitrary fiber degree, we prove that if a Higgs sheaf has reduced Friedman spectral curve, or is regular on a general fiber with non-reduced spectral cover, then its Higgs field takes values in the saturation of the pull-back of the canonical bundle of the base curve in the cotangent bundle of the surface. [ABSTRACT FROM AUTHOR]

Using deformation theory, we answer two questions of Shioda concerning the infinitesimal structure of the moduli space of integral sections in the Mordell–Weil group. [ABSTRACT FROM AUTHOR]

This article makes an important contribution to solving the long-standing problem of whether all elliptic curves can be equipped with a hash function (indifferentiable from a random oracle) whose running time amounts to one exponentiation in the basic finite field Fq{{\mathbb{F}}}_{q}. More precisely, we construct a new indifferentiable hash function to any ordinary elliptic Fq{{\mathbb{F}}}_{q}-curve Ea{E}_{a} of j-invariant 1728 with the cost of extracting one quartic root in Fq{{\mathbb{F}}}_{q}. As is known, the latter operation is equivalent to one exponentiation in finite fields with which we deal in practice. In comparison, the previous fastest random oracles to Ea{E}_{a} require to perform two exponentiations in Fq{{\mathbb{F}}}_{q}. Since it is highly unlikely that there is a hash function to an elliptic curve without any exponentiations at all (even if it is supersingular), the new result seems to be unimprovable.

Let S be a complex smooth projective surface of Kodaira dimension one. We show that the group Auts(S) of symplectic automorphisms acts trivially on the Albanese kernel CHo(S)alb of the 0-th Chow group CHo(S), unless possibly if the geometric genus and the irregularity satisfy pg(S) = q(S) ∈ {1, 2}. In the exceptional cases, the image of the homomorphism Auts(S) → Aut(CHo(S)alb) has the order at most 3. Our arguments actually take care of the group Autf(S) of fibration-preserving automorphisms of elliptic surfaces f: S → B. We prove that if σ ∈ Autf (S) induces the trivial action on Hi,0(S) for i > 0, then it induces the trivial action on CH0(S)alb. As a by-product we obtain that if S is an elliptic K3 surface, then Autf(S)∩Auts (S) acts trivially on CH0(S)alb. [ABSTRACT FROM AUTHOR]

We study triple covers of K3 surfaces, following Miranda (1985, American Journal of Mathematics 107, 1123–1158). We relate the geometry of the covering surfaces with the properties of both the branch locus and the Tschirnhausen vector bundle. In particular, we classify Galois triple covers computing numerical invariants of the covering surface and of its minimal model. We provide examples of non-Galois triple covers, both in the case in which the Tschirnhausen bundle splits into the sum of two line bundles and in the case in which it is an indecomposable rank 2 vector bundle. We provide a criterion to construct rank 2 vector bundles on a K3 surface S which determine a non-Galois triple cover of S. The examples presented are in any admissible Kodaira dimension, and in particular, we provide the constructions of irregular covers of K3 surfaces and of surfaces with geometrical genus equal to 2 whose transcendental Hodge structure splits in the sum of two Hodge structures of K3 type. [ABSTRACT FROM AUTHOR]

Let $(A,\mathfrak m)$ be an excellent two-dimensional normal local domain. In this paper, we study the elliptic and the strongly elliptic ideals of A with the aim to characterize elliptic and strongly elliptic singularities, according to the definitions given by Wagreich and Yau. In analogy with the rational singularities, in the main result, we characterize a strongly elliptic singularity in terms of the normal Hilbert coefficients of the integrally closed $\mathfrak m$ -primary ideals of A. Unlike $p_g$ -ideals, elliptic ideals and strongly elliptic ideals are not necessarily normal and necessary, and sufficient conditions for being normal are given. In the last section, we discuss the existence (and the effective construction) of strongly elliptic ideals in any two-dimensional normal local ring. [ABSTRACT FROM AUTHOR]

We show that, for each n > 0 , there is a family of elliptic surfaces which are covered by the square of a curve of genus 2 n + 1 , and whose Hodge structures have an action by Q (- n) . By considering the case n = 3 , we show that one particular family of K3 surfaces are covered by the squares of curves of genus 7. Using this, we construct a correspondence between the square of a curve of genus 7 and a general K3 surface in P 4 with 15 ordinary double points up to a map of finite degree of K3 surfaces. This gives an explicit proof of the Kuga–Satake–Deligne correspondence for these K3 surfaces and any K3 surfaces related to them by maps of finite degree, and further, a proof of the Hodge conjecture for the squares of these surfaces. We conclude that the motives of these surfaces are Kimura-finite. Our analysis gives a birational equivalence between a moduli space of curves with additional data and the moduli space of these K3 surfaces with a specific elliptic fibration. [ABSTRACT FROM AUTHOR]

We analyze Higgs bundles (V, ϕ) on a class of elliptic surfaces π : X → B, whose underlying vector bundle V has vertical determinant and is fiberwise semistable. We prove that if the spectral curve of V is reduced, then the Higgs field ϕ is vertical, while if the bundle V is fiberwise regular with reduced (respectively, integral) spectral curve, and if its rank and second Chern number satisfy an inequality involving the genus of B and the degree of the fundamental line bundle of π (respectively, if the fundamental line bundle is sufficiently ample), then ϕ is scalar. We apply these results to the problem of characterizing slope-semistable Higgs bundles with vanishing discriminant on the class of elliptic surfaces considered, in terms of the semistability of their pull-backs via maps from arbitrary (smooth, irreducible, complete) curves to X. [ABSTRACT FROM AUTHOR]

Let F q be a finite field and E b : y 2 = x 3 + b be an ordinary (i.e., non-supersingular) elliptic curve (of j-invariant 0) such that b ∈ F q and q ≢ 1 (mod 27) . For example, these conditions are fulfilled for the curve BLS12-381 ( b = 4 ). It is a de facto standard in the real-world pairing-based cryptography at the moment. This article provides a new constant-time hash function H : { 0 , 1 } ∗ → E b (F q) indifferentiable from a random oracle. Its main advantage is the fact that H computes only one exponentiation in F q . In comparison, the previous fastest constant-time indifferentiable hash functions to E b (F q) compute two exponentiations in F q . In particular, applying H to the widely used BLS multi-signature with m different messages, the verifier should perform only m exponentiations rather than 2m ones during the hashing phase. [ABSTRACT FROM AUTHOR]

We explore Bott Vanishing for elliptic surfaces over P 1 . We show that Bott Vanishing is significantly affected by the geometric properties of fibers. For example, whether there exist certain types of singular fibers on the elliptic fibration such as cuspidal fibers. For an ample line bundle on the surface with large self-intersection, these geometric properties give criteria for Bott Vanishing. [ABSTRACT FROM AUTHOR]

We construct simply connected, minimal, symplectic 4-manifolds with exotic smooth structures and each with one Seiberg–Witten basic class up to sign, on the Noether line and between the Noether and half Noether lines by star surgeries introduced by Karakurt and Starkston, and by using complex singularities. We also construct certain configurations of complex singularities in the rational elliptic surfaces geometrically, without using any monodromy arguments. By using these configurations, we give symplectic embeddings of star shaped plumbings inside (some blow-ups of) elliptic surfaces. [ABSTRACT FROM AUTHOR]

We construct geometric isogenies between three types of two-parameter families of K3 surfaces of Picard rank 18. One is the family of Kummer surfaces associated with Jacobians of genus-two curves admitting an elliptic involution, another is the family of Kummer surfaces associated with the product of two non-isogenous elliptic curves, and the third is the twisted Legendre pencil. The isogenies imply the existence of algebraic correspondences between these K3 surfaces and prove that the associated four-dimensional Galois representations are isomorphic. We also apply our result to several subfamilies of Picard rank 19. The result generalizes work of van Geemen and Top (Bull Lond Math Soc 38(2):209–223, 2006). [ABSTRACT FROM AUTHOR]

We introduce Kummer surfaces X = Km (C × C) with the group scheme G = μ 2 acting on the self-product of the rational cuspidal curve in characteristic two. The resulting quotients are normal surfaces having a configuration of sixteen rational double points of type A 1 , together with a rational double point of type D 4 . We show that our Kummer surfaces are precisely the supersingular K3 surfaces with Artin invariant σ ≤ 3 , and characterize them by the existence of a certain configuration of thirty curves. After contracting suitable curves, they also appear as normal K3-like coverings for simply-connected Enriques surfaces. [ABSTRACT FROM AUTHOR]

We study the moduli spaces of elliptic K3 surfaces of Picard number at least 3, that is, U⊕⟨−2k⟩‐polarized K3 surfaces. Such moduli spaces are proved to be of general type for k⩾220. The proof relies on the low‐weight cusp form trick developed by Gritsenko, Hulek and Sankaran. Furthermore, explicit geometric constructions of some elliptic K3 surfaces lead to the unirationality of these moduli spaces for k<11 and for 19 other isolated values up to k=64. [ABSTRACT FROM AUTHOR]

Given d ∈ N , we prove that any polarized Enriques surface (over any field k of characteristic p ≠ 2 or with a smooth K3 cover) of degree greater than 12 d 2 contains at most 12 rational curves of degree at most d. For d > 2 , we construct examples of Enriques surfaces of high degree that contain exactly 12 rational degree-d curves. [ABSTRACT FROM AUTHOR]

We describe the image of the locus of hyperelliptic curves of genus g under the period mapping in a neighbourhood of the diagonal locus Diagg. There is just one branch for each of the alkanes CgH2g+2 of elementary organic chemistry, and each branch has a simple linear description in terms of the entries of the period matrix. This picture is replicated for simply connected Jacobian elliptic surfaces, which form the next simplest class of algebraic surfaces after K3 and abelian surfaces. In the period domain for such surfaces of geometric genus g, there is a locus W1g that is analogous to Diagg, and the image of the moduli space under the period map has just one branch through W1g for each alkane. Each branch is smooth and has an explicit description as a vector bundle of rank g−1 over a domain that contains W1g. [ABSTRACT FROM AUTHOR]