Differentielle Signatur/Positiv/Charakteristik 0/Graduiert/Folgerungen/Fakt/en

Let ${}R$ be a standard-graded normal domain over an algebraically closed field ${}K$ of characteristic ${}0$. Suppose that ${}R$ is Gorenstein and that it has an isolated singularity. Suppose that the differential signature of ${}R$ is positive. Then the following hold.

1. The ${}a$-invariant of ${}R$ is negative.
2. ${}R$ has a rational singularity.
3. ${}R$ is log-terminal.
4. The Kodaira dimension of ${}\operatorname {Proj} R$ is negative.
5. ${}\operatorname {Proj} R$ is Fano.