## 代数幾何学セミナー

開催情報 火曜日　15:30～17:00　数理科学研究科棟(駒場) 122号室 權業 善範・中村 勇哉・田中公

### 2018年11月20日(火)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Artin-Mazur height, Yobuko height and
Hodge-Wittt cohomologies

[ 講演概要 ]
A few years ago Yobuko has introduced the notion of
a delicate invariant for a proper smooth scheme over a perfect field $k$
of finite characteristic. (We call this invariant Yobuko height.)
This generalize the notion of the F-splitness due to Mehta-Srinivas.

In this talk we give relations between Artin-Mazur heights
and Yobuko heights. We also give a finiteness result on
Hodge-Witt cohomologies of a proper smooth scheme $X$ over $k$
with finite Yobuko height. If time permits, we give a cofinite type result on
the $p$-primary torsion part of Chow group of of $X$
of codimension 2 if $\dim X=3$.

### 2018年11月27日(火)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Frobenius summands and the finite F-representation type (TBA)
[ 講演概要 ]
We are motivated by a question arising from commutative algebra, asking what kind of
graded rings in positive characteristic p have finite F-representation type. In geometric
setting, this is related to the problem to looking out for Frobenius summands. Namely,
given aline bundle L on a projective variety X, we want to know how many and what
kind of indecomposable direct summands appear in the direct sum decomposition of
the iterated Frobenius push-forwards of L. We will consider the problem in the following
two cases, although the present situation in (2) is far from satisfactory.
(1) two-dimensional normal graded rings (a joint work with Ryo Ohkawa)
(2) the anti-canonical ring of a quintic del Pezzo surface

### 2019年01月29日(火)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Logarithmic good reduction and the index (TBA)
[ 講演概要 ]
A proper smooth variety over a complete discrete valuation field is said to have (log) good reduction if it admits a proper (log) smooth model over the valuation ring (the log structure is given by the closed fiber). Monodromy criteria for good reduction and log good reduction have been studied. We study the log case by additional other conditions on geometric invariants such as the index of the variety (the minimal positive degree of a 0-cycle). In particular, we obtain a criterion for log good reduction of curves of genus one.