Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $x \in U$ be a germ of a $3$-dimensional terminal quotient singularity of type $\frac{1}{r} (1,a,r-a)$ over $\mathbb{C}$. Let $D \in |-K_U|$ be an anticanonical divisor. Assume that $D$ is normal and singular at $x$. Let $\mu: \tilde{U} \rightarrow (U,D)$ be a log resolution such that $\mu$ is isomorphism outside $x$.

Question Is $H^1(\tilde{D}, \mathcal{O}_{\tilde{D}}(-\tilde{D}) =0$?

It seems that $H^1(\tilde{U}, \mathcal{O}_{\tilde{U}}(-\tilde{D}))=0$.

If $H^2(\tilde{U}, \mathcal{O}_{\tilde{U}}(-2\tilde{D})) =0$, the statement in the Question seems to be true. Is there a counterexample?

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.