# Generalizing the Cross-ratio: The Moduli Space of N Points on the Projective <br/><br/>Line is Cut Out by Simple Quadrics if N is Not Six

Monday, September 18, 2006 - 9:30am - 10:20am

EE/CS 3-180

Ravi Vakil (Stanford University)

The cross-ratio is a classical gadget that let's you see if two

sets of four

points on the projective line are projectively equivalent

—

it is the

moduli space of four points on the projective line. The

generalization of

this to an arbitrary number of points leads to the notion of

the moduli space

of n points on the projective line, which is a projective

variety, one of the

most classical examples of a Geometric Invariant Theory

quotient. It also may

be interpreted as the space of all polygons in 3-space. We

show that this

space is actually cut out by quadrics, of a particularly simple

sort. This

talk is intended for a broad audience. (This is joint work

with B. Howard, J.

Millson, and A. Snowden, and deals with preprint

math.AG/0607372.)

sets of four

points on the projective line are projectively equivalent

—

it is the

moduli space of four points on the projective line. The

generalization of

this to an arbitrary number of points leads to the notion of

the moduli space

of n points on the projective line, which is a projective

variety, one of the

most classical examples of a Geometric Invariant Theory

quotient. It also may

be interpreted as the space of all polygons in 3-space. We

show that this

space is actually cut out by quadrics, of a particularly simple

sort. This

talk is intended for a broad audience. (This is joint work

with B. Howard, J.

Millson, and A. Snowden, and deals with preprint

math.AG/0607372.)