## Algebraic Geometry Seminar

Seminar information archive ～06/14｜Next seminar｜Future seminars 06/15～

Date, time & place | Friday 13:30 - 15:00 ハイブリッド開催/117Room #ハイブリッド開催/117 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | GONGYO Yoshinori, NAKAMURA Yusuke, TANAKA Hiromu |

### 2016/07/25

13:30-15:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Birational rigidity of complete intersections (English)

**Fumiaki Suzuki**(Tokyo)Birational rigidity of complete intersections (English)

[ Abstract ]

A complete intersection defined by s hypersurfaces of degree d_1, ... ,d_s in a projective space P^N is Q-Fano, i.e. normal, Q-factorial, terminal and having an ample anti-canonical divisor, if d_1 + ... + d_s is at most N and it has only mild singularities. Then it is rationally-connected by the results of Kollar-Miyaoka-Mori, Zhang and Hacon-Mckernan. A natural question is to determine its rationality. If its dimension or degree is at most 2, then it is rational. How about the remaining cases?

When d_1 + ... + d_s = N, birational rigidity give one of the most effective ways to tackle this problem. We recall that a Q-Fano variety is birationally superrigid if any birational map to the source of another Mori fiber space is isomorphism. It implies that X is non-rational and Bir(X) = Aut(X). After the works of Iskovskih-Manin, Pukhlikov, Chelt'so and de Fernex-Ein-Mustata, de Fernex proved that every smooth hypersurface of degree N in P^N is birationally superrigid for N at least 4. He also proved birational superrigidity of a large class of singular hypersurfaces of this type.

In this talk, we would like to extend de Fernex's results to complete intersections. As a key step, we generalize Pukhlikov's multiplicity bounds of cycles in hypersurfaces to complete intersections.

A complete intersection defined by s hypersurfaces of degree d_1, ... ,d_s in a projective space P^N is Q-Fano, i.e. normal, Q-factorial, terminal and having an ample anti-canonical divisor, if d_1 + ... + d_s is at most N and it has only mild singularities. Then it is rationally-connected by the results of Kollar-Miyaoka-Mori, Zhang and Hacon-Mckernan. A natural question is to determine its rationality. If its dimension or degree is at most 2, then it is rational. How about the remaining cases?

When d_1 + ... + d_s = N, birational rigidity give one of the most effective ways to tackle this problem. We recall that a Q-Fano variety is birationally superrigid if any birational map to the source of another Mori fiber space is isomorphism. It implies that X is non-rational and Bir(X) = Aut(X). After the works of Iskovskih-Manin, Pukhlikov, Chelt'so and de Fernex-Ein-Mustata, de Fernex proved that every smooth hypersurface of degree N in P^N is birationally superrigid for N at least 4. He also proved birational superrigidity of a large class of singular hypersurfaces of this type.

In this talk, we would like to extend de Fernex's results to complete intersections. As a key step, we generalize Pukhlikov's multiplicity bounds of cycles in hypersurfaces to complete intersections.