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## The index of an algebraic variety

### Dino Lorenzini

U. Georgia

###
Thursday, October 13, 2011

007 Kemeny Hall, 4 pm

Tea 3:30 pm, 300 Kemeny Hall

**Abstract: **
Let *K* be a field. Suppose that the algebraic variety is given by the set
of common solutions to
a system of polynomials in *n* variables with coefficients in *K*.
Given a solution *P*=(*a*_{1},...,*a*_{n}) of this system
with coordinates in the algebraic closure of *K* we associate to it an integer
called the degree of *P* and defined to be the degree of the extension
*K*(*a*_{1},...,*a*_{n}) over *K*.
When all coordinates *a*_{i} belong to
*K*, *P* is called a *K*-rational point, and its degree is 1.
The index of the variety is the greatest common divisor of all possible degrees of
points on *P*. It is clear that if there exists a
*K*-rational point on the variety, then the index equals 1.
The converse is not true in general. We
shall discuss in this talk various properties of the index.

This talk will be accessible to graduate students.