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.

For a nonconstant elliptic surface over $\mathbb {P}^1$ defined over $\mathbb {Q}$, it is a result of Silverman [‘Heights and the specialization map for families of abelian varieties’, J. reine angew. Math. 342 (1983), 197–211] that the Mordell–Weil rank of the fibres is at least the rank of the group of sections, up to finitely many fibres. If the elliptic surface is nonisotrivial, one expects that this bound is an equality for infinitely many fibres, although no example is known unconditionally. Under the Bunyakovsky conjecture, such an example has been constructed by Neumann [‘Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten. I’, Math. Nachr. 49 (1971), 107–123] and Setzer [‘Elliptic curves of prime conductor’, J. Lond. Math. Soc. (2) 10 (1975), 367–378]. In this note, we show that the Legendre elliptic surface has the desired property, conditional on the existence of infinitely many Mersenne primes.

In this note, we prove the semiampleness conjecture for Kawamata log terminal Calabi–Yau (CY) surface pairs over an excellent base ring. As applications, we deduce that generalized abundance and Serrano’s conjecture hold for surfaces. Finally, we study the semiampleness conjecture for CY threefolds over a mixed characteristic DVR.

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.

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.

We compute the nef cone of the Hilbert scheme of points on a general rational elliptic surface. As a consequence of our computation, we show that the Morrison–Kawamata cone conjecture holds for these nef cones.

The main goal of this paper is to show that Castelnuovo–Enriques’ $P_{12}$- theorem (a precise version of the rough classification of algebraic surfaces) also holds for algebraic surfaces $S$ defined over an algebraically closed field $k$ of positive characteristic ($\text{char}(k)=p>0$). The result relies on a main theorem describing the growth of the plurigenera for properly elliptic or properly quasielliptic surfaces (surfaces with Kodaira dimension equal to 1). We also discuss the limit cases, i.e., the families of surfaces which show that the result of the main theorem is sharp.

Let $k$ be a field of characteristic $2$. We give a geometric proof that there are no smooth quartic surfaces $S\subset \mathbb{P}_{k}^{3}$ with more than 64 lines (predating work of Degtyarev which improves this bound to 60). We also exhibit a smooth quartic containing 60 lines which thus attains the record in characteristic $2$.

This paper consists mainly of a review and applications of our old results relating to the title. We discuss how many elliptic fibrations and elliptic fibrations with infinite automorphism groups (or Mordell–Weil groups) an algebraic K3 surface over an algebraically closed field can have. As examples of applications of the same ideas, we also consider K3 surfaces with exotic structures: with a finite number of non-singular rational curves, with a finite number of Enriques involutions, and with naturally arithmetic automorphism groups.

We study the deformation theory of elliptic fiber bundles over curves in positive characteristics. As applications, we give examples of nonliftable elliptic surfaces in characteristics 2 and 3, which answer a question of Katsura and Ueno. Also, we construct a class of elliptic fibrations, whose liftability is equivalent to a conjecture of Oort concerning the liftability of automorphisms of curves. Finally, we classify deformations of bielliptic surfaces.

We prove that the elliptic surface ${{y}^{2}}={{x}^{3}}+2\left( {{t}^{8}}+14{{t}^{4}}+1 \right)x+4{{t}^{2}}\left( {{t}^{8}}+6{{t}^{4}}+1 \right)$ has geometric Mordell–Weil rank 15. This completes a list of Kuwata, who gave explicit examples of elliptic $K3$-surfaces with geometric Mordell–Weil ranks 0, 1, … , 14, 16, 17, 18.

A complex projective manifold is said to be weakly special if none of its finite étale covers has a dominant rational map to a positive-dimensional manifold of general type. Weakly special manifolds are conjectured by Harris and Tschinkel (Rational points on quartics, Duke Math. J. 104 (2000), 477–500) to be potentially dense if defined over a number field. In a previous paper by F. Campana (Special varieties, orbifolds and classification theory, Ann. Inst. Fourier (Grenoble) 54 (2004), 499–665), the stronger notion of special (complex projective) manifolds was introduced. These are conjectured in (9.20) of that work to be exactly the potentially dense manifolds (when defined over a number field). The two notions of specialness and weak-specialness coincide up to dimension 2, but differ from dimension 3 on, as shown by examples $X$ constructed by Bogomolov and Tschinkel (Special elliptic fibrations, in Proc. Fanu Conf., Torino, 2003, ed. A. Conte), answering a question raised in Campana's paper. These examples should thus be potentially dense, according to the first conjecture, but should not be so, according to the second conjecture. The present techniques of arithmetic geometry do not seem to be able to decide between these two conflicting conjectures. So, instead of this, we investigate the more tractable hyperbolic aspects of the examples of Bogomolov and Tschinkel, and show that for certain manifolds $X$, the behaviour of entire transcendental curves is that given in Conjecture 9.16, p. 618 of Campana's above-mentioned work. More precisely, the image of any holomorphic map from ${\mathbb{C}}$ to $X$ is contained either in some fibre of the unique elliptic fibration on $X$, or in some fixed divisor of $X$. This is consistent with the standard links (Lang and Vojta for varieties of general type, the first author's previous work in general) expected between the hyperbolic and arithmetic behaviour of projective varieties in cases where the second conjecture (stated by Campana) holds. Conversely, if the conjecture stated by Harris and Tschinkel is true, these expected links between arithmetics and complex hyperbolicity should fail to hold on these examples already.

Silverman has discussed the problem of bounding the Mordell–Weil ranks of elliptic curves over towers of function fields (J. Algebraic Geom. 9 (2000), 301–308; J. reine. angew. Math. 577 (2004), 153–169). We first prove generalizations of the theorems of Silverman by a different method, allowing non-abelian Galois groups and removing the dependence on Tate's conjectures. We then prove some theorems about the growth of Mordell–Weil ranks in towers of function fields whose Galois groups are $p$-adic Lie groups; a natural question is whether the Mordell–Weil rank is bounded in such a tower. We give some Galois-theoretic criteria which guarantee that certain curves ${\mathcal{E}}/{\mathbb{Q}}(t)$ have finite Mordell–Weil rank over ${\mathbb{C}}(t^{p^{-\infty}})$, and show that these criteria are met for elliptic K3 surfaces whose associated Galois representations have sufficiently large image.

In a recent paper Ahlgren, Ono and Penniston described the L-series of K3 surfaces from a certain one-parameter family in terms of those of a particular family of elliptic curves. The Tate conjecture predicts the existence of a correspondence between these K3 surfaces and certain Kummer surfaces related to these elliptic curves. A geometric construction of this correspondence is given here, using results of D. Morrison on Nikulin involutions.

In this paper, we study the modularity of certain elliptic surfaces by determining their $L$-series through their monodromy groups.

A Hopf surface is the quotient of the complex surface ${{\mathbb{C}}^{2}}\,\backslash \,\left\{ 0 \right\}$ by an infinite cyclic group of dilations of ${{\mathbb{C}}^{2}}$. In this paper, we study the moduli spaces ${{\mathcal{M}}^{n}}$ of stable $\text{SL}\left( 2,\,\mathbb{C} \right)$-bundles on a Hopf surface $\mathcal{H}$, from the point of view of symplectic geometry. An important point is that the surface $\mathcal{H}$ is an elliptic fibration, which implies that a vector bundle on $\mathcal{H}$ can be considered as a family of vector bundles over an elliptic curve. We define a map $G:\,{{\mathcal{M}}^{n}}\,\to \,{{\mathbb{P}}^{2n+1}}$ that associates to every bundle on $\mathcal{H}$ a divisor, called the graph of the bundle, which encodes the isomorphism class of the bundle over each elliptic curve. We then prove that the map $G$ is an algebraically completely integrable Hamiltonian system, with respect to a given Poisson structure on ${{\mathcal{M}}^{n}}$. We also give an explicit description of the fibres of the integrable system. This example is interesting for several reasons; in particular, since the Hopf surface is not Kähler, it is an elliptic fibration that does not admit a section.

We show that the use of orbifold bundles enables some questions to be reduced to the case of flat bundles. The identification of moduli spaces of certain parabolic bundles over elliptic surfaces is achieved using this method.

In this paper, we shall give a method in constructing dihedral Galois covering with prescribed branch locus. As an application, we shall look into dihedral Galois covering of P2, where torsion elements of the Mordell-Weil group of an elliptic surface play key roles in constructing coverings