Related papers: On the first Gaussian map for Prym-canonical line …

We show that the second fundamental form of the Prym map lifts the second gaussian map of the Prym-canonical bundle. We prove, by degeneration to binary curves, that this gaussian map is surjective for the general point [C,A] of R_g for g >…

We prove that, for the general curve of genus g, the 2nd Gaussian map is injective if g <= 17 and surjective if g >= 18. The proof relies on the study of the limit of the 2nd Gaussian map when the general curve of genus g degenerates to a…

By a theorem of Wahl, for canonically embedded curves which are hyperplane sections of K3 surfaces, the first gaussian map is not surjective. In this paper we prove that if C is a general hyperplane section of high genus (greater than 280)…

In this paper we prove that the Prym map, from the space of double coverings of a curve of genus g with r branch points to the moduli space of abelian varieties, is generically injective if r>6 and g>1, r=6 and g>2, r=4 and g>4, r=2 and…

Let C be a generic non-singular curve of genus g defined over a field of characteristic different from 2. We show that for every line bundle on C of degree at most g+1, the natural product map S^2(H^0(L))\to H^0(C,L^2) is injective. We also…

In this paper we consider the Prym map for double coverings of curves of genus $g$ ramified at $r>0$ points. That is, the map associating to a double ramified covering its Prym variety. The generic Torelli theorem states that the Prym map…

Let $C$ be a Petri general curve of genus $g$ and $E$ a general stable vector bundle of rank $r$ and slope $g-1$ over $C$ with $h^0 (C, E) = r+1$. For $g > (2r+2)(2r+1)$, we show how the bundle $E$ can be recovered from the tangent cone to…

We prove that the moduli space of double covers ramified at two points $\mathcal{R}_{g,2}$ is uniruled for $3\leq g\leq 6$ and of general type for $g\geq 16$. Furthermore, we consider Prym-canonical divisorial strata in the moduli space…

We prove the generic injectivity of the Prym map, sending a double covering of an elliptic curve ramified at r>4 points to its polarized Prym variety. For r=6 the map is birational.

We prove that the locus of Prym curves $(C,\eta)$ of genus $g \geq 5$ for which the Prym-canonical system $|\omega_C(\eta)|$ is base point free but the Prym--canonical map is not an embedding is irreducible and unirational of dimension…

We prove that the Prym map corresponding to \'etale cyclic coverings of hyperelliptic curves is injective whenever the degree of the covering $d \geq 6$ is not a power of an odd prime. For other degrees $d\geq 9$, we show that the Prym map…

We prove the Bertram-Feinberg-Mukai conjecture for a generic curve $C$ of genus $g$ and a semistable vector bundle $E$ of rank two and determinant $K$ on $C$, namely we prove the injectivity of the Petri-canonical map $S^2(H^0(E))\to…

In this paper we show a relation between higher even Gaussian maps of the canonical bundle on a smooth projective curve of genus $g \geq 4$ and the second fundamental form of the Torelli map. This generalises a result obtained by Colombo,…

In this work we use the bigonal, trigonal and tetragonal constructions to describe the fibers of the Prym map P : R_{g} ---->A_{g-1} inthe cases when it is dominant, i.e. for g < 7. The most interesting cases are g = 5, where the fiber is a…

Let C be a generic curve, E a generic vector bundle on C. Then, for every line bundle on C the twisted Petri map P:H^0(C,L\otimes E)\otimes H^0(C, K\otimes L^*\otimes E^{*})--> H^0(C, K) is injective.

In this paper we study Koszul cohomology and the Green and Prym-Green conjectures for canonical and Prym-canonical binary curves. We prove that if property $N_p$ holds for a canonical or a Prym-canonical binary curve of genus $g$ then it…

Green's Conjecture states the following : syzygies of the canonical model of a curve are simple up to the p^th stage if and only if the Clifford index of C is greater than p. We prove that the generic curve of genus g satisfies Green's…

In this paper we study higher even Gaussian maps of the canonical bundle for cyclic trigonal curves. More precisely, we study suitable restrictions of these maps determining a lower bound for the rank, and more generally, a lower bound for…

Let $C$ be a general canonical curve of genus $g$ defined over an algebraically closed field of arbitrary characteristic. We prove that if $g \notin \{4,6\}$, then the normal bundle of $C$ is semistable. In particular, if $g \equiv 1$ or…

We prove that the $k$-th Gaussian map $\gamma^k_{H}$ is surjective on a polarized unnodal Enriques surface $(S, H)$ with $\phi(H)>2k+4$. In particular, as a consequence, when $\phi(H)>4(k+2)$, we obtain the surjectivity of the $k$-th…