On the dimension of varieties of special divisors

Abstract:

Let ${T_g}$ denote the Teichmüller space and let $V$ denote the universal family of Teichmüller surfaces of genus $g$ Let $V_{{T_g}}^{(n)}$ denote the $n$th symmetric product of $V$ over ${T_g}$ and let $J$ denote the family of Jacobians over ${T_g}$. Let $f:V_{{T_g}}^{(n)} \to \text {J}$ be the natural relativization over ${T_g}$ of the classical map defined by integrating holomorphic differentials. Let \[ u:{f^\ast }\Omega _{\text {J} /{T_g}}^1 \to \Omega _{V_{{T_g}/{T_g}}^{(n)}}^1\] be the map induced by $f$. We define $G_n^r$ to be the analytic subspace of $V_{{T_g}}^{(n)}$ defined by the vanishing of ${ \wedge ^{n - r + 1}}u$. Put $\tau = (r + 1)(n - r) - rg$. We show that $G_n^1 - G_n^2$, if nonempty, is smooth of pure dimension $3g - 3 + \tau + 1$. From this result, we may conclude that, for a generic curve $X$, the fiber of $G_n^1 - G_n^2$ over the module point of $X$, if nonempty, is smooth of pure dimension $\tau + 1$, a classical assertion. Variational formulas due to Schiffer and Spencer and Rauch are employed in the study of $G_n^r$.