通常の講演時間はおよそ 1 時間半です。
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.
$N$ 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。