Quadrik/Charakteristik 0/Syzygienbündel/Gradschranke/Fakt/en

Let ${}K$ be an algebraically closed field of characteristic zero. Let ${}f$ be an irreducible quadric in ${}d+1$ variables, ${}d\geq 2$.

Then the symmetric powers ${}\operatorname {Sym} ^{n}(\operatorname {Syz} (x_{1},\ldots ,x_{d+1}))(m)$ on the quadric

${}Q_{d}=\operatorname {Proj} K[x_{1},\ldots ,x_{d+1}]/(f)\,$

have no nontrivial section for ${}m<{\frac {3}{2}}n$.