Room P3.10, Mathematics Building

Bruno Oliveira, University of Miami
Twisted symmetric differentials and quadric envelopes

Subvarieties of projective space of low codimension have no symmetric differentials. If we twist the cotangent bundle by multiples of the polarization, \(\Omega^1_X\otimes O(a)\), one still has no sections of its symmetric powers if $a\lt 1$ (result of Schneider 92).

The case $a=1$ is the interesting border case, sections might exist and we call these sections twisted symmetric differentials. We will give a geometric description of the space of twisted symmetric differentials and show that if the codimension of the subvariety $X$ of $P^n$ is small relative to its dimension, $\operatorname{cod}(X)\lt \dim(X)/2$, then the algebra of twisted symmetric differentials of $X$ is the algebra generated by the quadrics containing $X$ (it will be shown for $\operatorname{cod}(X)=2$ and the general case will be discussed).