arXiv subjects

Luca Cesarano

Publications and source records attributed to Luca Cesarano.

4 recordsLinked to original sources

arXiv2020

Canonical Maps of general Hypersurfaces in Abelian Varieties

The main theorem of this paper is that, for a general pair $(A,X)$ of an (ample) Hypersurface $X$ in an Abelian Variety $A$, the canonical map $\Phi_X$ of $X$ is birational onto its image if the polarization given by $X$ is not principal (i.e., its Pfaffian $d$ is not equal to $1$). We also show that, setting $g = dim (A)$, and letting $d$ be the Pfaffian of the polarization given by $X$, then if $X$ is smooth and $$\Phi_X : X \rightarrow \mathbb{P}^{N:=g+d-2}$$ is an embedding, then necessarily we have the inequality $ d \geq g + 1$, equivalent to $N : = g+d-2 \geq 2 \ dim(X) + 1.$ We also formulate the following interesting conjecture, motivated by work of the second author: if $ d \geq g + 1,$ then, for a general pair $(A,X)$, $\Phi_X$ is an embedding.

math.AG↗

arXiv2019

Biquadratic addition laws on elliptic curves in $\mathbb{P}^3$ and the canonical map of the $(1,2,2)$-Theta divisor

We recall that a smooth ample surface $\mathcal{S}$ in a general $(1,2,2)$-polarized abelian threefold, which is the pullback of the Theta divisor of a smooth plane quartic curve $\mathcal{D}$, is a surface isogenous to the product $\mathcal{C} \times \mathcal{C}$, where $\mathcal{C}$ is a genus $9$ curve embedded in $\mathbb{P}^3$ as complete intersection of a smooth quadric and a smooth quartic. We show that the space of global holomorhic sections of the canonical bundle of this surface is generated by certain determinantal bihomogeneous polynomials of bidegree $(2,2)$ on $\mathbb{P}^3$, which can be used to define biquadratic addition laws on the Jacobi model of elliptic curves, embedded in $\mathbb{P}^3$ as complete intersection of two quadrics. Finally, we use this interesting relationship with the biquadratic addition laws to describe the behavior of the canonical map of $\mathcal{S}$.

math.AG↗

arXiv2019

On birational trivial families and Adjoint quadrics

Let $\pi\colon \mathcal{X}\to B$ be a family whose general fiber $X_b$ gives a $(d_1,...,d_a)$ polarisation of a general Abelian variety where $1\leq d_i\leq 2$, $i=1,...,a$ and $a\geq 4$. We show that the fibers are in the same birational class if all the $(m,0)$ forms on $X_b$ are liftable to $(m,0)$ forms on $\mathcal{X}$ where $m=1$ and $m=a-1$. Actually we show general criteria to find families with fibers in the same birational class, which leads together with a famous theorem of Nori to some interesting applications.

math.AG↗

arXiv2018

Very ampleness of the canonical bundle of surfaces of type (1, 2, 2) on abelian threefolds

The present work deals with the canonical map of smooth, compact complex surfaces of general type in a polarization of type $(1,2,2)$ on an abelian threefold. A natural and classical question is whether the canonical system of such surfaces is very ample in the general case. In this work, we provide a positive answer to this question. First, we describe the structure of the canonical map of those smooth ample surfaces of type $(1,2,2)$ in an abelian threefold which are bidouble cover of principal polarizations. Then, we study the general behavior of the canonical map of general ample surfaces $\mathcal{S}$, yielding a $(1,2,2)$-polarization on an abelian threefold $A$ which is isogenous to a product. By combining these descriptions, we show that the canonical map yields a holomorphic embedding when $A$ and $\mathcal{S}$ are both sufficiently general. It follows, in particular, a proof of the existence of canonical irregular surfaces in $\mathbb{P}^5$ with numerical invariants $p_g = 6$, $q = 3$ and $K^2 = 24$.

math.AG↗