ncku-talk2018-12-21

Colloquium

DATE 2018-12-21　11:10-12:00

PLACE 數學館3176教室

SPEAKER Yusuke Nakamura (中村勇哉) 助理教授（University of Tokyo）

TITLE A Rational Point Problem on Fano Varieties

ABSTRACT Fano varieties defined over a certain field $k$ are believed to have a $k$-rational point. For instance, the classically known Chevalley--Warning theorem states that homogeneous polynomials $f_1,\ldots, f_l \in \mathbb{F} _q [x_0, \ldots, x_n]$ with $(n+1)$-many variables over a finite field $\mathbb{F} _q$ have a non-trivial common solution if $\sum _{1 \le i \le l} \deg f_i \le n$ holds. More precisely, it states that the number of common solution is divisible by the characteristic $p$ of the field $\mathbb{F} _q$. Geometrically, it can be interpreted as that the number of the $\mathbb{F} _q$-rational points on a complete intersection Fano variety is one modulo $p$. This kind of problem was answered in the smooth case by Esnault. She proved that smooth Fano varieties defined over a finite field have a rational point. It is natural to ask whether we can generalize this results to singular Fano varieties or not. In this talk, we study the rational point problem on singular Fano varieties and the relation to the minimal model program.

Colloquium

DATE	2018-12-21　11:10-12:00

PLACE	數學館3176教室

SPEAKER	Yusuke Nakamura (中村勇哉) 助理教授（University of Tokyo）

TITLE	A Rational Point Problem on Fano Varieties

ABSTRACT	Fano varieties defined over a certain field $k$ are believed to have a $k$-rational point. For instance, the classically known Chevalley--Warning theorem states that homogeneous polynomials $f_1,\ldots, f_l \in \mathbb{F} _q [x_0, \ldots, x_n]$ with $(n+1)$-many variables over a finite field $\mathbb{F} _q$ have a non-trivial common solution if $\sum _{1 \le i \le l} \deg f_i \le n$ holds. More precisely, it states that the number of common solution is divisible by the characteristic $p$ of the field $\mathbb{F} _q$. Geometrically, it can be interpreted as that the number of the $\mathbb{F} _q$-rational points on a complete intersection Fano variety is one modulo $p$. This kind of problem was answered in the smooth case by Esnault. She proved that smooth Fano varieties defined over a finite field have a rational point. It is natural to ask whether we can generalize this results to singular Fano varieties or not. In this talk, we study the rational point problem on singular Fano varieties and the relation to the minimal model program.