Tangential Alexander polynomials and non-reduced degeneration

A combinatorial approach to Alexander-Hirschowitz Theorem based on toric degenerations

We present an alternative proof of the Alexander-Hirschowitz Theorem in dimension 3 using degenerations of toric varieties.

A new proof of the Alexander-Hirschowitz interpolation Theorem

The classical polynomial interpolation problem in several variables can be generalized to the case of points with greater multiplicities. What is known, as yet, is essentially concentrated in the Alexander-Hirschowitz Theorem which says that a general collection of double points in P^r gives independent conditions on the linear system L of the hypersurfaces of degree d, with a well known list of exceptions. We present a new proof of this theorem which consists in performing degenerations of P^r and analyzing how L degenerates.

Twisted Alexander polynomials of Plane Algebraic Curves

We consider the Alexander polynomial of a plane algebraic curve twisted by a linear representation. We show that it divides the product of the polynomials of the singularity links, for unitary representations. Moreover, their quotient is given by the determinant of its Blanchfield intersection form. Specializing in the classical case, this gives a geometrical interpretation of Libgober's divisibility Theorem. We calculate twisted polynomials for some algebraic curves and show how they can detect Zariski pairs of equivalent Alexander polynomials and that they are sensitive to nodal degeneration…

关于圆锥曲线-直线排列的Alexander多项式

On the Alexander polynomials of conic-line arrangements

本文计算了与束相关的圆锥曲线-直线排列的Alexander多项式，证明了Halphen束的一般性结果，并构造了具有7次单位根的非约化排列，填补了该类多项式的空白。