Local properties of secant varieties in symmetric products. I

Abstract:

Let $L$ be a line bundle on an abstract nonsingular curve $C$, let $V \subset {H^0}(C,L)$ be a linear system, and denote by ${C^{(d)}}$ the symmetric product of $d$ copies of $C$. There exists a canonically defined ${C^{(d)}}$-bundle map: \[ \sigma :V \otimes {\mathcal {O}_{{C^{(d)}}}} \to {E_L},\] where ${E_L}$ is a bundle of rank $d$ obtained from $L$ by a so-called symmetrization process. The various degenerary loci of $\sigma$ can be considered as subsecant schemes of ${C^{(d)}}$. Our main result, Theorem 4.2, is given in $\S 4$, where we obtain a local matrix description of $\sigma$ valid (also) at points on the diagonal in ${C^{(d)}}$, and thereby we can determine the completions of the local rings of the secant schemes at arbitrary points. In $\S 5$ we handle the special case of giving a local scheme structure to the zero set of $\sigma$.