5
$\begingroup$

A projective normal and $\mathbb{Q}$-factorial variety $X$ is said to be log Fano if there exists and effective divisor $D$ on $X$ such $-K_X-D$ is ample and the pair $(X,D)$ is klt.

Now, let $f:X\dashrightarrow Y$ birational map which is an isomorphism in codimension one between two projective normal and $\mathbb{Q}$-factorial varieties. Assume that the strict transform of $-K_X$ on $Y$ is nef and big. Can we conclude that then $X$ is log Fano? I know that this holds if the strict transform of $-K_X$ on $Y$ is ample.

Thank you very much.

$\endgroup$
5
$\begingroup$

You need to know that $Y$ as at worst klt singularities. In this case $Y$ is a weak Fano variety with klt singularities and then it is log Fano.

Now, you are done since a small $\mathbb{Q}$-factorial transfomation of a log Fano variety is log Fano.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.