相关论文: On the extendability of elliptic surfaces of rank …

We study Mordell-Weil rank jumps on families of jacobians of a pencil of genus-2 curves on a K3 surface defined over a number field k. We exhibit a finite extension l/k over which the subset of fibers for which the rank jumps is infinite.…

We compute some numerical invariants of the lines on hyperplane sections of a smooth cubic threefold over complex numbers. We also prove that for any smooth hypersurface $X\subset \mathbb P^{n+1}$ of degree $d$ over an algebraically closed…

Let $X$ be a K3 surface over a number field. We prove that $X$ has infinitely many specializations where its Picard rank jumps, hence extending our previous work with Shankar--Shankar--Tang to the case where $X$ might have potentially bad…

We discuss the geometry of the genus one fibrations associated to an elliptic fibration on a K3 surface. We show that the two-torsion subgroup of the Brauer group of a general elliptic fibration is naturally isomorphic to the two-torsion of…

Using toric geometry, lattice theory, and elliptic surface techniques, we compute the Picard Lattice of certain K3 surfaces. In particular, we examine the generic member of each of M. Reid's list of 95 families of Gorenstein K3 surfaces…

This article provides, over any field, infinitely many algebraic embeddings of the affine spaces $\mathbb{A}^1$ and $\mathbb{A}^2$ into smooth quadrics of dimension two and three respectively, which are pairwise non-equivalent under…

We show several examples of integrable systems related to special K3 and rational surfaces (e.g., an elliptic K3 surface, a K3 surface given by a double covering of the projective plane, a rational elliptic surface, etc.). The construction,…

We prove the filling area conjecture in the hyperelliptic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu's result in genus 0. We translate the problem into a question about closed…

Atiyah classifies vector bundles on elliptic curves $E$ over an algebraically closed field of any characteristic. On the other hand, a rank $2$ vector bundle on $E$ defines a surface $S$ with a $\mathbb{P}^1$-bundle structure on $E$. We…

We prove an extension criterion for codimension one foliations on projective hypersurfaces based on the degree of the foliation and the degree of the hypersurface, and we ensure, in some instances, an isomorphism between the corresponding…

We show that there exists a quasi-isometric embedding of the product of $n$ copies of $\mathbb{H}_{\mathbb{R}}^2$ into any symmetric space of non-compact type of rank $n$, and there exists a bi-Lipschitz embedding of the product of $n$…

We study holomorphic symplectic manifolds which are fibred by abelian varieties. This structure is a higher dimensional analogue of an elliptic fibration on a K3 surface. We investigate when a holomorphic symplectic manifold is fibred in…

Using some theory of (rational) elliptic surfaces plus elementary properties of a Mordell-Weil group regarded as module over the endomorphism ring of a (CM) elliptic curve, we present examples of such surfaces with j-invariant zero. In…

We present a detailed study of elliptic fibrations on Fourier-Mukai partners of K3 surfaces, which we call derived elliptic structures. We fully classify derived elliptic structures in terms of Hodge-theoretic data, similar to the Derived…

Nishiyama introduced a lattice theoretic classification of the elliptic fibrations on a $K3$ surface. In a previous paper we used his method to exhibit $52$ elliptic fibrations, up to isomorphisms, of the singular $K3$ surface of…

We solve the problem of counting jacobian elliptic fibrations on an arbitrary complex projective K3 surface up to automorphisms. We then illustrate our method with several explicit examples.

Let $\f: X \ra Z$ be a proper surjective map from a smooth complex manifold $X$ onto a normal variety $Z$. If $\f$ has connected fibers and $-K_X$ is $\f$-ample then $\f$ is called a good contraction. In the present paper we study good…

In this paper we give a complete local parametric classification of the hypersurfaces with dimension at least three of a space form that carry a totally geodesic foliation of codimension one. A classification under the assumption that the…

We investigate the geometric properties of marginally trapped surfaces (surfaces which have null mean curvature vector) in the spaces of oriented geodesics of Euclidean 3-space and hyperbolic 3-space, endowed with their canonical neutral…

The paper presents a generalized Weierstrass representation for pseudospherical surfaces in terms of 3x3 matrices, using moving frames and loop group decompositions. The construction of all such surfaces, starting from a given…