Let X be a sufficiently general smooth k-gonal curve of genus g and R ∈ Pic(X) the degree k spanned line bundle. We find an optimal integer z > 0 such that the line bundle $R^{\otimes z}$ is very ample and projectively normal.

The classical Segre theory gives a necessary and sufficient condition for a plane curve to be a branch curve of a (generic) projection of a smooth surface in ${\mathbb{P}}^3$. We generalize this result for smooth surfaces in a projective space of any dimension in the following way: given two plane curves, $B$ and $E$, we give a necessary and sufficient condition for $B$ to be the branch curve of a surface $X$ in ${\mathbb{P}}^N$ and $E$ to be the image of the double curve of a ${\mathbb{P}}^3$-model of $X$. In the classical Segre theory, a plane curve...

Here we study the integers (d, g, r) such that on a smooth projective curve of genus g there exists a rank r stable vector bundle with degree d and spanned by its global sections.

Let C = (C, g^1/4 ) be a tetragonal curve. We consider the scrollar invariants e1 , e2 , e3 of g^1/4 . We prove that if W^1/4 (C) is a non-singular variety, then every g^1/4 ∈ W^1/4 (C) has the same scrollar invariants.

Let $Y$ be an integral projective curve with $g:=p_a\left(Y\right)\ge 2$. For all positive integers $d$, $r$ let $W_d^r\left(Y\right)\left({}^{*A*}\right)$ be the set of all $L\in {\text{Pic}}^d\left(Y\right)$ with $h^0\left(Y,L\right)\ge r+1$ and $L$ spanned. Here we prove that if $d\le g-2$, then $dim\left(W_d^r\left(Y\right)\left({}^{*A*}\right)\right)\le d-3r$ except in a few cases (essentially if $Y$ is a double covering).

Sia $X\subset \mathbb{P}\left(H^0{\left(X,L\right)}^{*}\right)$ una curva proeittiva e lissa, generali nel senso di Brill-Noether, indichiamo con ${\mathcal{R}}_4\left(X\right)$ l'insieme algebrico di quadrici di rango $4$ contenendo a $X$. In questo lavoro noi descriviamo birazionalmente i componenti irriducibile di ${\mathcal{R}}_4\left(X\right)$.

A lattice model with exponential interaction, was proposed and integrated by M. Toda in the 1960s; it was then extensively studied as one of the completely integrable (differential-difference) equations by algebro-geometric methods, which produced both quasi-periodic solutions in terms of theta functions of hyperelliptic curves and periodic solutions defined on suitable Jacobians by the Lax-pair method. In this work, we revisit Toda’s original approach to give solutions of the Toda lattice in terms...

We consider the linear system $|2{\Theta }_0|$ of second order theta functions over the Jacobian $JC$ of a non-hyperelliptic curve $C$. A result by J.Fay says that a divisor $D\in |2{\Theta }_0|$ contains the origin $𝒪\in JC$ with multiplicity $4$ if and only if $D$ contains the surface $C-C=\left\{𝒪\right(p-q)\mid p,q\in C\}\subset JC$. In this paper we generalize Fay’s result and some previous work by R.C.Gunning. More precisely, we describe the relationship between divisors containing $𝒪$ with multiplicity $6$, divisors containing the fourfold $C_2-C_2=\{𝒪(p+q-r-s)\mid p,q,r,s\in C\}$, and divisors singular along $C-C$, using the third exterior...

We study the Torelli morphism from the moduli space of stable curves to the moduli space of principally polarized stable semi-abelic pairs. We give two characterizations of its fibers, describe its injectivity locus, and give a sharp upper bound on the cardinality of finite fibers. We also bound the dimension of infinite fibers.