Seminar Algebra and Geometry: Kevin Langlois (Institut Fourier, Grenoble)

Integral closure and affine varieties with a torus action

Let A=C[f₁,..., f_r] be an integral algebra of finite type over the field of complex numbers.  Using the elements f₁,..,f_r 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^*×...×C^*=(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 theM-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.

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