Picard bundles and Twisted Picard bundles on the Jacobian of a curve

Let $Y$ denote an irreducible projective curve with at most nodes as singularities and defined over an algebraically closed field of characteristic zero. We study the restriction of the twisted Picard bundles on the compactified Jacobian $\bar{J}(Y)$ of $Y$ to the embedded curve in $\bar{J}(Y)$. As an application, we show that for $g=2$ and each integer $r\ge 3$, there is a two-dimensional family of stable ACM bundles on the compactified Jacobian which has the Picard bundle in its limit. We define an embedding ${\alpha }_Y$ of the (generalised) Jacobian $J(Y)$ in the moduli space $U_Y^s(n,d)$ of stable vector bundles on $Y$ using a twisted restriction $E_Y$ of a Picard bundle to $Y\subset J(Y)$. We show that (under suitable conditions) the restriction of the universal bundle $\mathcal{U}$ to $Y\times J(Y)$ is stable for suitable polarisation. For the embedding of a smooth curve $Y$ given by $E_Y\otimes B,B$ a line bundle of degree $b$, we show that the restriction of the Picard bundle on $U_Y^s(n,d+nb)$ to $J(Y)$ is $\theta$-semistable for $b\ge 2g-1$ and $\theta$-stable for $b\ge 2g$. We also determine the relation between the restriction of the theta divisor on $U_Y^s(n,d+nb)$ to $J(Y)$ and the theta divisor $\theta$ on $\bar{J}(Y)$.