Заседание Математического семинара Глобус 22 апреля 2010 года

Автор темы Даниил Кальченко 
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Заседание Математического семинара Глобус 22 апреля 2010 года
В четверг, 22 апреля 2010 года, в 15:40 в конференц-зале НМУ, Б. Власьевский 11, состоится доклад «On beauville surfaces and the genus of the curves arising in their construction». Докладчик – Gabino Gonzalez-Diez (Universidad Autonoma de Madrid).

A (unmixed) Beauville surface is a complex surface of the form $S=(C_1\times C_2)/G$ where $C_1$ and $C_2$ are complex curves of genus $\geq 2$ and $G$ is a finite group acting freely on the product $C1\times C2$ in such a way that each of the factors is preserved by the action and, moreover, the quotient $C_i/G$ is an orbifold of genus zero with three cone points. Beauville surfaces were introduced by Catanese following an initial construction of Beauville (of a surface of general type with invariants $p_g=q=0$) in which $C_1=C_2$ is the Fermat curve $X_0^5+X_1^5+X_2^5=0$ and $G\simeq\mathbb{Z}/5\mathbb{Z}\times\mathbb{Z}/5\mathbb{Z}$.

Striking properties of Beauville surfaces are 1) despite of being of general type they are rigid and 2) the curves $C_i$ and the group $G$ are determined by $S$. (For instance these two properties imply that these surfaces tend to possess Galois conjugates non homeomorphic to themselves)

In this talk I shall discuss questions such as which groups $G$ and which genera $g_1\leq g_2$ of $C1, C2$ can arise in the construction of Beauville surfaces. In particular I will show that $g_1$ and $g_2$ have to be $\geq 6$ and that if $g_1=6$ then $S$ agrees with (one of the two) Beauville examples above. The proof of this fact will rely on methods belonging to the theory of Riemann surfaces.

Математический семинар Глобус
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