Local properties of secant varieties in symmetric products. II

Abstract:

Let $V$ be a linear system on a curve $C$. In Part I we described a method for studying the secant varieties $V_d^r$ locally. The varieties $V_d^r$ are contained in the $d$-fold symmetric product ${C^{(d)}}$. In this paper (Part II) we apply the methods from Part I. We give a formula for local tangent space dimensions of the varieties $V_d^1$ valid in all characteristics (Theorem 2.4). Assume $\operatorname {rk}\;V = n + 1$ and $\operatorname {char} K = 0$. In $\S \S 3$ and $4$ we describe in detail the projectivized tangent cones of the varieties $V_n^1$ for a large class of points. The description is a generalization of earlier work on trisecants for a space curve. In $\S 5$ we study the curve in ${C^{(2)}}$ consisting of divisors $D$ such that $2D \in V_4^1$ . We give multiplicity formulas for all points on this curve in ${C^{(2)}}$ in terms of local geometrical invariants of $C$. We assume $\operatorname {char} K = 0$.