Seminar Algebra & Geometry: Kevin Langlois
Kevin Langlois, Institut Fourier, Grenoble
Integral closure and affine varieties with a torus action
Let A=C[f1,..., fr] be an integral algebra of finite type over the field of complex numbers. Using the elements f1,..,fr it is difficult in general to describe the normalization of A.
In this talk, we provide some examples whenever A is a multigraded algebra.
Consider the group T=C*x...xC*=(C*)n given by the componentwise multiplication. We say that T is an algebraic torus of dimension n. Let M be the character lattice
of T. Then a T-action on X=Spec A is equivalent to endow A with a M-graduation.
We classify the M-graded algebras A by a number called complexity. Geometrically, it corresponds to the codimension of general T-orbits in X. Algebraically, the complexity is somehow "the thickness of graded pieces" of the algebra A.
The problem of normalization for complexity zero case is well known (monomial or toric case). For the complexity one, the normalization of A admits a construction due to Timashev and Altmann-Hausen in terms of polyhedral divisors over an algebraic smooth curve. Taking homogeneous generators, we will explain how to build the polyhedral divisor corresponding to the normalization of A.
Assume that A is normal. Then A is given by a polyhedral divisor. A similar problem arises for the integral closure of homogeneous ideals. We will give an answer for the
complexity one case. We will provide also a classification of homogeneous integrally
closed ideals of A.