Reposted by サッカー bones
サッカー bones
@doctorbones.bsky.social
557 followers 166 following 2.4K posts
Mathematical skeleton. More Frog than Bird. Big chilling with my homies. Bad at math and worse at physics. Acolyte of the hypergeometric. Aviation enjoyer. I’m the cookie. 我非常喜欢数学。
![Screenshot of LaTeX editor:
(begin (proposition}
(labelfprop-nonempty}
[Prokhorox, Icite[Thm. 1.15]{MR4922010}]
Suppose that $V$ is a Fano $n$-fold of index $d \geq n-2$. Then $|-K_V| Ineq lemptyset$.
lend(proposition}
응
At present, the general non-emptiness of $|-K_VI$ for higher dimensional Fano varieties of low index is not known to the best of our knowledge.
This leads us to the following critical assumption that we use throughout the remainder of the article.
%
(begin (assumption)
(label fass-nonempty}
Let $V$ be an almost minimal Fano $n$-fold.
Then $|-K_V| \neq lemptyset$, and a general, smooth, irreducible divisor $X^\vee lin I-K_V|$ is a CY variety.
lend {assumption}
응
\par In the context of this article, a CY xacietx of dimension $m$ is a smooth xaciety of dimension with trivial canonical class and
$h^{i,0}=0$ for $i = 1,\dots, m-1$. Such a variety
may be closed (resp. open), and hence projective (resp. quasiprojective). Given our assumptions, associated to a Fano $n$-fold $V$ are two natural CY varieties of respective dimension $n-1, l, n+1$:
$X^\vee lin |-K_V|$, a very general smooth](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:26ypzvcklxb4w5qnyrr7h5cu/bafkreifrdqodi2lofzsjxl32wzpinmbhznf2j6ludbcmftydrpwmpwon4u)
