Curves and Surfaces

Friday, April 13, 2007 - 11:00am - 12:30pm
EE/CS 3-180
Ragni Piene (University of Oslo)
This talk is intended as a brief introduction to certain aspects of the
theory of algebraic curves and surfaces.

To a projective algebraic variety one associates its arithmetic genus,
using the constant term of the Hilbert polynomial. Since a complex
projective algebraic curve can be viewed as a compact Riemann surface, it
also has a topological genus, equal to the number of holes in the Riemann
surface. Hirzebruch's Riemann-Roch theorem says that the topological genus
is equal to the arithmetic genus. I sketch a proof of this fundamental
result, using algebra, geometry, and topology. For algebraic surfaces
there is a corresponding result, Noether's formula, expressing the
arithmetic genus in terms of topological invariants, which can be proved
in a similar way. The last part of the talk discusses the theory of curves
on surfaces, in particular the links to classical enumerative geometry and
to modern string theory in theoretical physics.