Derivatives of Wronskians with applications to families of special Weierstrass points

Abstract:

Let $\pi : \mathcal {X} \longrightarrow S$ be a flat proper family of smooth connected projective curves parametrized by some smooth scheme of finite type over $\mathbb {C}$. On every such a family, suitable derivatives “along the fibers” (in the sense of Lax) of the relative wronskian, as defined by Laksov and Thorup, are constructed. They are sections of suitable jets extensions of the $g(g+1)/2$-th tensor power of the relative canonical bundle of the family itself. The geometrical meaning of such sections is discussed: the zero schemes of the $(k-1)$-th derivative ($k\geq 1$) of a relative wronskian correspond to families of Weierstrass Points (WP’s) having weight at least $k$. The locus in $M_{g}$, the coarse moduli space of smooth projective curves of genus $g$, of curves possessing a WP of weight at least $k$, is denoted by $wt(k)$. The fact that $wt(2)$ has the expected dimension for all $g\geq 2$ was implicitly known in the literature. The main result of this paper hence consists in showing that $wt(3)$ has the expected dimension for all $g\geq 4$. As an application we compute the codimension $2$ Chow ($Q$-)class of $wt(3)$ for all $g\geq 4$, the main ingredient being the definition of the $k$-th derivative of a relative wronskian, which is the crucial tool which the paper is built on. In the concluding remarks we show how this result may be used to get relations among some codimension $2$ Chow ($Q$-)classes in $M_{4}$ ($g\geq 4$), corresponding to varieties of curves having a point $P$ with a suitable prescribed Weierstrass Gap Sequence, relating to previous work of Lax.