Birationally rigid Fano complete intersections. II

✍ Scribed by Pukhlikov, Aleksandr V.

• Book ID: 121873037
• Publisher: Walter de Gruyter GmbH & Co. KG
• Year: 2014
• Tongue: English
• Weight: 252 KB
• Volume: 2014
• Category: Article
• ISSN: 0075-4102
• DOI: 10.1515/crelle-2012-0059

✦ Synopsis

Abstract.

We prove that a generic (in the sense of Zariski topology) Fano complete
intersection V of the type

(

d
1

,
⋯
,

d
k

)

$(d_1,\dots ,d_k)$

in

ℙ

M
+
k

${\mathbb {P}}^{M+k}$

, where

d
1

⋯
+

d
k

=
M
+
k

$d_1+\dots +d_k=M+k$

, is birationally superrigid
if

M
≥
7

$M\ge 7$

,

M
≥
k
+
3

$M\ge k+3$

and

max
{

d
i

}
≥
4

$\operatorname{max} \lbrace d_i\rbrace \ge 4$

. In
particular, on the variety V there is exactly one structure of a
Mori fibre space (or a rationally connected fibre space), the
groups of birational and biregular self-maps coincide,

Bir
V

Aut
V

$\operatorname{Bir} V= \operatorname{Aut} V$

, and the variety V is
non-rational. This fact covers a considerably larger range of
complete intersections than our result of
[J. reine angew. Math 541 (2001), 55–79],
which required the condition

M
≥
2
k
+
1

$M\ge 2k+1$

.