広島大学理学部数学科 代数数理グループ

2025年度の代数学セミナー

通常の講演時間はおよそ 1 時間半です。

今後の予定

922(月) 1500 分   於 広島大学理学部 A 棟 A202 号室 (曜日・開始時間にご注意ください)
岡 睦雄 氏(東京科学大学) (15:00-16:00)
Almost Newton non-degenerate functions and some application
In this talk, we introduce a class ”almost Newton non-degenerate functions” of functions which contains Newton non-degenerate functions and shows some applications. Roughly speaking, it is a class of functions with an isolated singularity at the origin and it can be resolved by hight two toric modifications (a tree with finite branches). We give a formula for the zeta functions which generalize Varchenko’ formula for the Newton non-degenerate functions.

Nguyen Thi Ngoc Giao 氏(東京理科大学) (16:30-17:30)
The lengths of plane Cremona transformations
We work over an algebraically closed field of characteristic zero, such as the field $\mathbb{C}$ of complex numbers. Our focus is on birational self-maps of the projective plane, also known as plane Cremona transformations.

First, we will recall the fundamental relations among the degree and the multiplicities of the base points of a plane Cremona transformation, that are encoded in the so-called homaloidal type of the map. We will see how to distinguish proper and improper homaloidal types by using Hudson's test.

Next we will recall that plane Cremona transformations of fixed degree $d$ form a quasi-projective variety $\Bir_d$ and we will see that irreducible components of $\Bir_d$ are in one-to-one correspondence with proper homaloidal types. We will also see that a few properties of $\Bir_d$ are known and we will address some open problem concerning them, like the connectedness.

A prominent role among plane Cremona transformations is played by de Jonquières maps, which are those of degree $d$ with a base point of multiplicity $d-1$. In particular, de Jonquières maps are the key ingredient needed to define the length of a plane Cremona transformation according to Blanc and Furter.

We will see how to refine this notion by introducing both the quadratic length and the ordinary quadratic length of a plane Cremona transformation. Then, we will show the known results about these lengths for plane Cremona transformations of small degree. Finally, we will discuss how to generalize this approach to give lower and upper bounds for both the quadratic length and the ordinary quadratic length of de Jonquières maps in any degree, and similarly for other homaloidal types.

The last part is a joint work in progress with Alberto Calabri.

これまでの記録

730(水) 1500 分   於 広島大学理学部 A 棟 A202 号室 (開始時間にご注意ください)
柳原 亮祐 氏(東北大学)
捻じれFermat商曲線の数論幾何とFleck数について
代数体上の非特異射影的代数曲線 $C$ のルートナンバー $w$ とは $C$ の完備L函数 $\Lambda(s,C)$ の函数等式に現れる符号 $w=\pm 1$ の事でParity conjecture($C$ のJacobi多様体のMordell-Weil rankの偶奇と $w$ が対応する)に代表されるように数論的に意義深いデータを持っており重要な研究対象とされている。

$N$ を正の整数、$\delta\neq 0$ を $\mathrm{ord}_{\ell}(\delta)=0$ or $\ell\nmid\mathrm{ord}_{\ell}(\delta)$ を満たす $\ell^N$-th power freeな整数とする。

この講演では $X^{\ell^N}+Y^{\ell^N}=\delta$ の商曲線の場合にルートナンバーを計算することが出来たのでその結果について述べる。この結果はStoll (2002), Shu (2021)の拡張にあたる。

またその過程でHilbert記号の特殊値として組み合わせ論で長い歴史を持つFleck数が現れる事を見出したのでそれについても述べる。