2015-2016 Schedule
Unless otherwise stated, all meetings in the Spring 2016 semester will be held in on Fridays from 2:00-4:00 PM in Room 5212.
February 19th, 2016
Richard Gustavson
Projective Varieties and Projective Closure
In this talk, we introduce projective varieties, which allow us to fill many "holes" that are present in affine varieties. We will begin by stating many alternative definitions of projective varieties, and then we will discuss the projective analogues of many of the results from affine algebraic geometry. In the projective case, the vanishing ideals of varieties are homogeneous ideals, and this presents certain complications. This discussion leads to the proof of the projective Nullstellensatz. We finish the talk by discussing the homogenization of a multivariate polynomial and the projective closure of an affine variety.
February 26th, 2016
Eli Amzallag
Projective Elimination Theory
Now that we have introduced projective varieties and done some work in describing them, we wish to carry out an investigation similar to the one we did for affine varieties. We will see that results parallel to the Elimination and Extension Theorems can be written down. One might expect that we could just somehow "homogenize" these results from the affine case to obtain the projective versions. However, some technical hurdles force us to work with different algebraic objects from the ones used in the affine formulations.
March 4th, 2016
Peter Thompson
Bezout's Theorem
We focus on curves in P^2(C), two-dimensional complex projective space. Our main result will be Bezout's Theorem: Let C and D be curves in P^2(C) with no common component. If C and D are defined by equations of degree m and n, respectively, then they intersect at exactly m*n points, counting multiplicities. Along the way, we will review some useful properties of resultants.
March 11th, 2016, 3:00-5:00 PM (Note the time change!)
Richard Gustavson
Varieties and Monomial Ideals
In this talk we begin our discussion of the dimension of a variety. As with previous topics, we want to give a definition that allows us to calculate the dimension of an arbitrary variety. We approach this by studying monomial ideals, for which there is a relatively simple method for calculating the dimension of the corresponding variety. In order to study arbitrary ideals, we need to examine the complement of a monomial ideal, that is, all of those monomials not in the given monomial ideal. In particular, we show that for s sufficiently large, the number of monomials not in a given monomial ideal of degree at most s is a polynomial in s. This polynomial, which eventually will be called the Hilbert polynomial, will allow us to define the dimension of the variety of an arbitrary ideal.
March 18th, 2016
Bin Guan
The Hilbert Function and the Dimension of a Variety
In this talk we will use the experience gained in the last talk to define the Hilbert function of an ideal I and use it to define the dimension of a variety V. We will give the definition in both the affine and projective cases. The basic idea will be to define the Hilbert function by the number of monomials not contained in the ideal I which are "linearly independent modulo" I. In the affine (resp. projective) case, we will use the number of monomials not in I of total degree at most (resp. equal to) s. For s sufficiently large, the Hilbert function is actually a polynomial in s, called the Hilbert polynomial of I. The dimension of a nonempty variety is defined by the degree of the Hilbert polynomial of its corresponding ideal. If time permits, we will state several basic properties of dimension, using the above definition.
April 1st, 2016
Bin Guan
Elementary Properties of Dimension
This talk will continue the last one. In the previous talk we defined the dimension of a variety by the degree of the Hilbert polynomial of the corresponding ideal. Using this definition we can now state several basic properties of dimensions. We will study the relation between the dimension of a variety and the number of defining equations; we will get a simple criterion for detecting when a variety has dimension 0; we will study the dimension of a union of varieties, and this property will allow us to reduce to the case of an irreducible variety when computing dimensions; finally we will show a property of dimensions of irreducible varieties.
April 15th, 2016
Peter Thompson
Dimension of an Affine Variety and Algebraic Independence
We have defined the dimension of an affine variety as the degree of its Hilbert polynomial, and have shown that this is equivalent to the dimension of the largest coordinate subspace of the vanishing set of the leading term ideal of the associated ideal. In this talk, we show that these two notions are equivalent to a third: the maximum number of algebraically independent elements of the coordinate ring. We also discuss a connection to Noether normalization.
May 6th, 2016
Eli Amzallag
Dimension and Nonsingularity
Intuition tells us that the more equations we have defining a variety, the smaller we should expect its dimension to be. In this talk, we explain how this intuition can be made precise for an affine variety, using the notion of nonsingularity and the Jacobian of the polynomials defining the variety. Although some proofs will only pertain to the case where the given variety is a hypersurface of
, the theorems we will state regarding dimension and nonsingularity hold for any affine variety.