On large theta-characteristics with prescribed vanishing

Let $C$ be a smooth projective curve of genus $g\geq 2$. Fix an integer $r\geq 0$, and let $\underline{k}=(k_1,\ldots,k_n)$ be a sequence of positive integers with $k_1+\ldots+k_n=g-1$. 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 such that $h^0\left(C,L\right)$ is at least $r+1$ and it has the same parity as $r+1$. 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 $r\geq 0$, $\underline{k}$ 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.

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