Related papers: Geometry of genus one fine compactified universal …

Let $X$ be a smooth geometrically connected projective curve over the field of fractions of a discrete valuation ring $R$, and $\mathfrak{m}$ a modulus on $X$, given by a closed subscheme of $X$ which is geometrically reduced. The…

In this article we provide a stack-theoretic framework to study the universal tropical Jacobian over the moduli space of tropical curves. We develop two approaches to the process of tropicalization of the universal compactified Jacobian…

We construct a birational model of the generalised Kummer fourfold of the Jacobian of a genus two curve, based on a geometric interpretation of the addition law on this Jacobian, obtained by the properties of the linear system of cubics on…

The Torelli theorem establishes that the Jacobian of a smooth projective curve, together with the polarization provided by the theta divisor, fully characterizes the curve. In the case of nodal curves, there exists a concept known as fine…

Let C be an integral projective curve with planar singularities. Consider its Jacobian J and the compactified Jacobian J'. We construct a flat family P of Cohen-Macaulay sheaves on J' parametrized by J'; over J, the family P is the Poincare…

We study certain moduli spaces of sheaves on Enriques surfaces thereby obtaining, in every odd dimension, new examples of Calabi-Yau manifolds. We describe the geometry (canonical bundle, fundamental group, second Betti number and certain…

We study the family of algebraic curves of genus $\geq 1$ defined by the affine equations $y^s=ax^r+b$ over a number field $k$, where $r \geq 2$ and $s\geq 2$ are fixed integers. Assuming the strong version of Lang's conjecture on varieties…

Given a finite, flat morphism between embeddable noetherian schemes of pure dimension 1, we define the notion of direct and inverse image for generalized divisors and generalized line bundles. In the case when we deal with (possibly…

In this paper, we study combinatorial properties of stable curves. To the dual graph of any nodal curve, it is naturally associated a group, which is the group of components of the N\'eron model of the generalized Jacobian of the curve. We…

We consider two cycles on the moduli space of compact type curves and prove that they coincide. The first is defined by pushing forward the virtual fundamental classes of spaces of relative stable maps to an unparameterized rational curve,…

Given a smooth curve with weighted marked points, the Abel-Jacboi map produces a line bundle on the curve. This map fails to extend to the full boundary of the moduli space of stable pointed curves. Using logarithmic and tropical geometry,…

A simply laced Dynkin diagram gives rise to a family of curves over $\mathbb{Q}$ and a coregular representation, using deformations of simple singularities and Vinberg theory respectively. Thorne has conjectured and partially proven a…

We generalize Macdonald's formula for the cohomology of Hilbert schemes of points on a curve from smooth curves to curves with planar singularities: we relate the cohomology of the Hilbert schemes to the cohomology of the compactified…

The moduli space $\cM_g$ of nonsingular projective curves of genus $g$ is compactified into the moduli $\bcM_g$ of Deligne-Mumford stable curves of genus $g$. We compactify in a similar way the moduli space of abelian varieties by adding…

Pagani and Tommasi have introduced a class of smoothable fine compactified Jacobians $\overline{\mathcal{J}}_{g,n}^d(\sigma)\rightarrow \overline{\mathcal{M}}_{g,n}$ over the moduli space of stable curves, depending nontrivially on the…

Using the compactified universal jacobian over the moduli space of stable marked curves, we give an expression in terms of natural classes of the zero section of the compactified universal jacobian the (rational) Chow ring. After extending…

There is a canonical identification, due to the author, of a convex real projective structure on an orientable surface of genus g and a pair consisting of a conformal structure together with a holomorphic cubic differential on the surface.…

We study smooth curves on abelian surfaces, especially for genus 4, when the complementary subvariety in the Jacobian is also a surface. We show that up to translation there is exactly one genus 4 hyperelliptic curve on a general (1,…

We consider the question: which elliptic curves appear as the Jacobian of a smooth curve of genus one splitting a Severi--Brauer variety? We provide three new examples. First, we show that if $E$ is any elliptic curve over an algebraically…

We observe that general reducible curves in sufficiently positive linear systems on K3 surfaces are of a form that generalises Kodaira's classification of singular elliptic fibres and thus call them extended ADE curves. On such a curve $C$,…