On large theta-characteristics with prescribed vanishing

Abstract

Let C be a smooth projective curve of genus $$g\ge 2$$ . Fix an integer $$r\ge 0$$ , and let $$\underline{k}=(k_1,\ldots ,k_n)$$ be a sequence of positive integers with $$\sum _{i=1}^n k_i =g-1$$ . In this paper, we study n-pointed curves $$(C,p_1,\ldots ,p_n)$$ such that the line bundle $$L:=O_C\left( \sum _{i=1}^n k_i p_i\right) $$ is a theta-characteristic with $$h^0\left( C,L\right) \ge r+1$$ and $$h^0\left( C,L\right) \equiv r+1 \,\mathrm {(mod\,2)}$$ . We prove that they describe a sublocus $${\mathcal {G}}^r_g(\underline{k})$$ of $${\mathcal {M}}_{g,n}$$ having codimension at most $$g-1+\frac{r(r-1)}{2}$$ . Moreover, for any $$r\ge 0$$ , $$\underline{k}$$ as above, and g greater than an explicit integer g(r) depending on r, we present irreducible components of $${\mathcal {G}}^r_g(\underline{k})$$ attaining the maximal codimension in $${\mathcal {M}}_{g,n}$$ , so that the bound turns out to be sharp.