Alladi Ramakrishnan Hall

Picard bundles and Brill-Noether loci in the compactified jacobian of a nodal curve

A. J. Parameswaran

TIFR, Mumbai

Let $Y$ denote an irreducible projective nodal curve with $k$ nodes and
of genus $g(Y)$. We prove a generalization of the classical Poincare formula to the
compactified Jacobian $\overline{J}(Y)$, the moduli space of torsionfree sheaves of
rank $1$, fixed degree $d$ on $Y$. We apply it to show that the Brill-Noether loci in
$\overline{J}(Y)$ are nonempty if the Brill-Noether number is nonnegative. We prove
that for $d\ge 2g(Y)$, the Picard bundle on $\overline{J}(Y)$ is stable but not ample,
unlike in the case of a smooth curve. However for the pull back of the
Picard bundle
to the desingularization of $\overline{J}(Y)$, the restriction to a general complete
intersection subvariety of codimension $k$ is ample. We use this to show that the
Brill-Noether loci are connected if the Brill-Noether number is bigger than $k$.
We prove that the Picard bundle is semistable for $d=2g(Y)-1$.