# 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.

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.