Date & Time: Wednesday, September 8, 2010, 15:30-16:30.
Venue: Ramanujan Hall
Title: Cyclic multiple planes
Speaker: Alok Maharana, TIFR
Abstract: Cyclic multiple planes are smooth affine complex algebraic surfaces defined by the equation $z^n=f(x,y)$ where $f(x,y)$ is a polynomial in two variables. Our talk consists of the following two parts. \\ \par
In the first part we shall show how to classify all cyclic multiple planes which are $\mathbb{Q}$-homology planes. Here a $\mathbb{Q}$-homology plane is by definition a smooth affine complex algebraic surface $X$ such that $H_i(X;\mathbb{Q})=0$ for $i\geq{1}$. Very few examples of hypersurface $\mathbb{Q}$-homology planes were known before and our classification provides many new examples of such hypersurfaces. It turns out that all such cyclic multiple planes have log-Kodaira dimension $\leq{1}$.\\ \par
In the second part we shall show how to classify all cyclic multiple planes with log-Kodaira dimension $\leq{1}$, i.e. without any hypothesis about their topology. We shall classify these surfaces with the additional hypothesis of non-vanishing irregularity if the log-Kodaira dimension of such a surface is zero.