On planar Cremona maps of prime order

p-elementary subgroups of the Cremona group of rank 3

For the subgroups of the Cremona group $\mathrm{Cr}_3(\mathbb C)$ having the form $(\boldsymbolμ_p)^s$, where $p$ is prime, we obtain an upper bound for $s$. Our bound is sharp if $p\ge 17$.

A Minkowski-style bound for the order of the finite subgroups of the Cremona group of rank 2 over an arbitrary field

Let Cr(k) be the Cremona group of rank 2 over a field k, i.e. the group of all k-automorphisms of k(X,Y). We determine the l.c.m. of the orders of the finite subgroups of Cr(k) of order prime to the characteristic of k.

Tori in the Cremona groups

We classify up to conjugacy the subgroups of certain types in the full, in the affine, and in the special affine Cremona groups. We prove that the normalizers of these subgroups are algebraic. As an application, we obtain new results in the Linearization Problem generalizing to disconnected groups Bia\lynicki-Birula's results of 1966-67. We prove "fusion theorems" for $n$-dimensional tori in the affine and in the special affine Cremona groups of rank $n$. In the final section we introduce and discuss the notions of Jordan decomposition and torsion primes for the Cremona groups.

Bijective Cremona transformations of the plane

We study the birational self-maps of the projective plane over finite fields that induce permutations on the set of rational points. As a main result, we prove that no odd permutation arises over a non-prime finite field of characteristic two, which completes the investigation initiated by Cantat about which permutations can be realized this way. Main ingredients in our proof include the invariance of parity under groupoid conjugations by birational maps, and a list of generators for the group of such maps.

平面克雷莫纳群的\(p\)-元非循环子群

$p$-elementary non-cyclic subgroups of the Cremona group of the plane

研究平面克雷莫纳群中同构于\((\mathbb{Z}/p\mathbb{Z})^r\)（\(p\)素数，\(r \geq 2\)）的子群，在特定代数闭域上进行共轭分类，给出代表列表和共轭性研究，还给出特征等于\(p\)时的相关结果。

假设Segre-Harbourne-Gimigliano-Hirschowitz猜想成立的平面线性系统有界性结果

Boundedness Results for Planar Linear Systems Assuming The Segre-Harbourne-Gimigliano-Hirschowitz Conjecture

本文在Segre-Harbourne-Gimigliano-Hirschowitz猜想成立的前提下，研究胀开$n\geq10$个一般点的射影平面上的完全线性系统，完成相关分类并证明有界性等结果。

Klein's ten planar dessins of degree 11, and beyond

We reinterpret ideas in Klein's paper on transformations of degree $11$ from the modern point of view of dessins d'enfants, and extend his results by considering dessins of type $(3,2,p)$ and degree $p$ or $p+1$, where $p$ is prime. In many cases we determine the passports and monodromy groups of these dessins, and in a few small cases we give drawings which are topologically (or, in certain examples, even geometrically) correct. We use the Bateman-Horn Conjecture and extensive computer searches to support a conjecture that there are infinitely many primes of the form $p=(q^n-1)/(q-1)$ for som…

本质p-维数与陈数

Essential p-dimension and Chern numbers

本文研究本质p-维数与陈数，证明有限p群在特定光滑射影簇上的作用可分解，推导对基域算术敏感的不动点定理，得到Cremona群循环p子群阶的界，关键利用Karpenko与Merkurjev的p群本质p-维数计算。