Restriction of the Poincar\'e bundle to a Calabi-Yau hypersurface

Let \cMx be the moduli space of stable vector bundles of rank $n\geq 3$ and determinant $\xi$ over a connected Riemann surface $X$, with $n$ and $d(\xi)$ coprime. Let $D$ be a Calabi-Yau hypersurface of \cMx. Denote by $U_D$ the restriction of the universal bundle to $X\times D$. It is shown that the restriction $(U_D)_x$ to $x\times D$ is stable, for any $x\in X$. Furthermore, for a general curve the connected component of the moduli space of semistable sheaves over $D$, containing $(U_D)_x$, is isomorphic to $X$. It is also shown that $U_D$ is stable for any polarisation, and the connected component of the moduli space of semistable sheaves over $X\times D$, containing $U_D$, is isomorphic to the Jacobian. Moreover, this is an isomorphism of polarised varieties, and hence such a moduli spaces determine the Reimann surface.Comment: AMSLaTex file. To appear in Crelles

On Chow Stability for algebraic curves

In the last decades there have been introduced different concepts of stability for projective varieties. In this paper we give a natural and intrinsic criterion of the Chow, and Hilbert, stability for complex irreducible smooth projective curves $C\subset \mathbb P ^n$. Namely, if the restriction $T\mathbb P_{|C} ^n$ of the tangent bundle of $\mathbb P ^n$ to $C$ is stable then $C\subset \mathbb P ^n$ is Chow stable, and hence Hilbert stable. We apply this criterion to describe a smooth open set of the irreducible component $Hilb^{P(t),s}_{{Ch}}$ of the Hilbert scheme of $\mathbb{P} ^n$ containing the generic smooth Chow-stable curve of genus $g$ and degree $d>g+n-\left\lfloor\frac{g}{n+1}\right\rfloor.$ Moreover, we describe the quotient stack of such curves. Similar results are obtained for the locus of Hilbert stable curves.Comment: Minor corrections and improvements to presentation. We add Theorem 4.

On coherent systems of type (n,d,n+1) on Petri curves

We study coherent systems of type $(n,d,n+1)$ on a Petri curve $X$ of genus $g\ge2$. We describe the geometry of the moduli space of such coherent systems for large values of the parameter $\alpha$. We determine the top critical value of $\alpha$ and show that the corresponding ``flip'' has positive codimension. We investigate also the non-emptiness of the moduli space for smaller values of $\alpha$, proving in many cases that the condition for non-emptiness is the same as for large $\alpha$. We give some detailed results for $g\le5$ and applications to higher rank Brill-Noether theory and the stability of kernels of evaluation maps, thus proving Butler's conjecture in some cases in which it was not previously known.Comment: 33 page