## The Brill-Noether Theorem for Real Algebraic Curves, by
Sharad Chaudhary

This thesis concerns real algebraic curves. Equivalently one may
consider compact Riemann surfaces together with an anti-holomorphic
involution. The basic problem is to find the minimal degree of a map
from a real algebraic curve to a genus zero curve. Motivation for
studying this problem comes on the one hand from the theory of
non-orientable minimal surfaces (soap films) and on the other hand from
classical algebraic geometry. This work focuses on the case in which
the curve has a real point-- equivalently, the anti-holomorphic
involution of the Riemann surface has a fixed point. In this case the
genus zero curve will always be one dimensional projective space.

The analogous problem for algebraic curves over the complex numbers
dates from about 1870. Brill and Noether proposed a solution which was
verified rigorously by a variety of different techniques between 1960 and
1985. The minimal degree of a map to a genus zero curve depends on
the choice of the curve. However, for "most" curves the answer is
always the same, namely [(g+3)/2], where g=genus and [] indicates the
integer part of a rational number. This number is
called the Brill-Noether number for the given genus. Every curve of a given
genus admits a map to a genus zero curve whose degree is less than
or equal to the Brill-Noether number.

The behaviour of real curves is more complicated. There exist real curves
of most topological types in genus 4 and 8 which do not admit any map
to a genus zero curve of degree less than or equal to the Brill-Noether number.
In fact this phenomenon occurs in an open subset of moduli in the
classical topology. On the other hand there exist other open
subsets in the same moduli spaces where the curves do admit maps to
genus zero curves whose degree is the Brill-Noether number.
The behaviour in genus 3, 5, 6, and 7 is quite different--here the
degree of the minimal map is always at most the Brill-Noether number.
In fact it is shown that this phenomenon holds for an infinite, albeit
rather sparse set of genuses. Precisely what happens in most high genuses
is still an open question.