Related papers: Generic Torelli and local Schottky theorems for Ja…

In [5], without giving a detailed proof, Yamauchi provided a formula to calculate the genus of a certain family of smooth complete intersection algebraic curves. That formula is used extensively in [1] to study the algebraic curves for…

We establish a generic vanishing theorem for surfaces in characteristic $p$ that lift to $W_2(k)$ and use it for surface classification of surfaces of general type with Euler characteristic 1 and large Albanese dimension.

Motivated by the recent surge of interest in the geometry of hybrid spaces, we prove an Abel-Jacobi theorem for a metrized complex of Riemann surfaces, generalizing both the classical Abel-Jacobi theorem and its tropical analogue.

In our PH.D. thesis we have showed that the Generalized Grothendieck's Conjecture of Periods applied to 1-motives, whose underlying semi-abelian variety is a product of elliptic curves and of tori, is equivalent to a transcendental…

The Jacobian elliptic functions are generalized to functions including the generalized trigonometric functions. The paper deals with the basis property of the sequence of generalized Jacobian elliptic functions in any Lebesgue space. In…

We consider generalizations of Szpiro's classical discriminant conjecture to hyperelliptic curves over a number field $K$, and to smooth, projective and geometrically connected curves $X$ over $K$ of genus at least one. The main results…

We prove an equivalence between the infinitesimal Torelli theorem for top forms on a hypersurface contained inside a Grassmannian $\mathbb G$ and the theory of adjoint volume forms presented in L. Rizzi, F. Zucconi, "Generalized adjoint…

It is known that in the case of hyperelliptic curves the Shafarevich conjecture can be made effective, i.e., for any number field k and any finite set of places S of k, one can effectively compute the set of isomorphism classes of…

Let $X$ be an elliptic K3 surface endowed with two distinct Jacobian elliptic fibrations $\pi_i$, $i=1,2$, defined over a number field $k$. We prove that there is an elliptic curve $C\subset X$ such that the generic rank over $k$ of $X$…

Adapting focal loci techniques used by Chiantini and Lopez, we provide lower bounds on the genera of curves contained in very general surfaces in Gorenstein toric threefolds. We illustrate the utility of these bounds by obtaining results on…

We prove that the geometric genus p of a curve in a very generic Jacobian of dimension g>3 satisfies either p=g or p>2g-3. This gives a positive answer to a conjecture of Naranjo and Pirola. For low values of g the second inequality can be…

We prove a local-global principle for torsors under the prosolvable geometric fundamental group of a hyperbolic curve over a number field.

We study the enumerative geometry of algebraic curves on abelian surfaces and threefolds. In the abelian surface case, the theory is parallel to the well-developed study of the reduced Gromov-Witten theory of K3 surfaces. We prove complete…

We show that a trivial case of Janelidze's categorical Galois theorem can be used as a key step in the proof of Joyal and Tierney's result on the representation of Grothendieck toposes as localic groupoids. We also show that this trivial…

We study the 2-parity conjecture for Jacobians of hyperelliptic curves over number fields. Under some mild assumptions on their reduction, we prove the conjecture over quadratic extensions of the base field. The proof proceeds via a…

We consider higher-dimensional analogues of the classical Brauer-Siegel theorem focusing on the case of abelian varieties over global function fields. We prove such an analogue in the case of constant families of elliptic curves and abelian…

Generalizations of the Jacobi and Weyl theorems on finite-dimensional linear flows to the case of linear flows on infinite-dimensional tori are presented. Conditions for periodicity, non-wandering, ergodicity and transitivity of…

It has long been known that to a complex cubic surface or threefold one can canonically associate a principally polarized abelian variety. We give a construction which works for cubics over an arithmetic base. This answers, away from the…

To a generic hypersurface in the affine torus $(\mathbb{C}^*)^n$ we associate a hypersurface arrangement in the projective space $\mathbb{P}^n$ consisting of the $n+1$ coordinate hyperplanes and a generic hypersurface, and compute the…

We construct algebraic curves in abelian surfaces starting from tropical curves in real tori. We give a necessary and sufficient condition for a tropical curve in a real torus to be realizable by an algebraic curve in an abelian surface.…