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
