Symmetric differentials on weighted hypersurfaces

Abstract

We study symmetric differentials on weighted hypersurfaces. Let $k$ be an algebraically closed field and let $$ X_d\subset \mathbb{P}_k(a_0,\ldots,a_N),\qquad N\ge3, $$ be a weighted hypersurface, with $a_{\min}:=\min_i a_i$. In characteristic zero we prove twisted vanishing in the range $t<m a_{\min}$ for quasi-smooth hypersurfaces of degree $d\ge2a_{\min}$, with the corresponding reflexive statement on the coarse space. We also prove a tame positive-characteristic analogue; in particular, reflexive symmetric differentials of every positive order vanish. In the second part we specialize to $k=\mathbb{C}$ and to surfaces in weighted projective three space. Using the quotient-singularity criterion of Asega--De Oliveira--Weiss together with weighted Kummer and Segre constructions, we produce singular weighted hypersurfaces whose minimal resolutions have big cotangent bundle. Combining the vanishing and bigness results with simultaneous resolution yields deformation-equivalent smooth projective surfaces for which the symmetric plurigenera jump from zero in every positive order to cubic asymptotic growth.