Related papers: Just-likely intersections on Hilbert modular surfa…

Let $A$ be a non-isotrivial ordinary abelian surface over a global function field with good reduction everywhere. Suppose that $A$ does not have real multiplication by any real quadratic field with discriminant a multiple of $p$. We prove…

We compute the generating series for the intersection pairings between the total Chern classes of the tangent bundles of the Hilbert schemes of points on a smooth projective surface and the Chern characters of tautological bundles over…

Let $D \le 2000$ be a positive discriminant such that $F = \mathbf{Q}(\sqrt{D})$ has narrow class one, and $A/F$ an abelian surface of ${\rm GL}_2$-type with everywhere good reduction. Assuming that $A$ is modular, we show that $A$ is…

Let S, T be surfaces in P3. Suppose that S intersect T is set-theoretically a smooth curve C of degree d and genus g. Suppose that S and T have no common singular points. Then if C is not a complete intersection, then deg(S), deg(T) < 2d^4.…

We call a log variety (X, D) algebraically hyperbolic if there exists a positive number e such that 2g(C) - 2 + i(C, D) >= e deg(C) for all curves C on X, where i(C, D) is the number of the intersections between D and the normalization of…

The paper is a contribution to the conjecture of Kobayashi that the complement of a generic curve in the projective plane is hyperbolic, provided the degree is at least five. Previously the authors treated the cases of two quadrics and a…

Principally polarized abelian surfaces with prescribed real multiplication (RM) are parametrized by certain Hilbert modular surfaces. Thus rational genus 2 curves correspond to rational points on the Hilbert modular surfaces via their…

Given a genus two curve $X: y^2 = x^5 + a x^3 + b x^2 + c x + d$, we give an explicit parametrization of all other such curves $Y$ with a specified symplectic isomorphism on three-torsion of Jacobians $\mbox{Jac}(X)[3] \cong…

We prove that any arithmetically Gorenstein curve on a smooth, general hypersurface $X\subset \bbP^{4}$ of degree at least 6, is a complete intersection. This gives a characterisation of complete intersection curves on general type…

We study relations in the Grothendieck ring of varieties which connect the Hilbert scheme of points on a cubic hypersurface $Y$ with a certain moduli space of twisted cubic curves on $Y$. These relations are generalizations of the…

We study arithmetic intersections on twisted (quaternionic) Hilbert modular surfaces and Shimura curves over a real quadratic field. Our first main result is the determination of the degree of the top arithmetic Todd class of an arithmetic…

Moduli spaces of hyperbolic surfaces with geodesic boundary components of fixed lengths may be endowed with a symplectic structure via the Weil-Petersson form. We show that, as the boundary lengths are sent to infinity, the Weil-Petersson…

Let $F$ be a non-singular homogeneous polynomial of degree $d$ in $n$ variables. We give an asymptotic formula of the pairs of integer points $(\mathbf x, \mathbf y)$ with $|\mathbf x| \le X$ and $|\mathbf y| \le Y$ which generate a line…

Let M be a compact, orientable, irreducible, atoroidal 3-manifold with boundary an incompressible torus. Techniques based on the characteristic submanifold theory are used to bound the intersection number of two slopes \alpha and \beta on…

For a large class of isotrivial rational elliptic surfaces (with section), we show that the set of rational points is dense for the Zariski topology, by carefully studying variations of root numbers among the fibers of these surfaces. We…

We study properties of rational curves on complete intersections in positive characteristic. It has long been known that in characteristic 0, smooth Calabi-Yau and general type varieties are not uniruled. In positive characteristic,…

We study a natural generalization of transversally intersecting smooth hypersurfaces in a complex manifold: hypersurfaces, whose components intersect in a transversal way but may be themselves singular. Such hypersurfaces will be called…

Due to the orbifold singularities, the intersection numbers on the moduli space of curves $\bar{\sM}_{g,n}$ are in general rational numbers rather than integers. We study the properties of the denominators of these intersection numbers and…

Let $C$ be a smooth curve. In this paper we investigate the geometric properties of the double nested Hilbert scheme of points on $C$, a moduli space introduced by the third author in the context of BPS invariants of local curves and sheaf…

We prove an unexpected general relation between the Jacobian syzygies of a projective hypersurface $V\subset \mathbb{P}^n$ with only isolated singularities and the nature of its singularities. This allows to establish a new method for the…