# GIT semistability of Hilbert points of Milnor algebras

##
Maksym Fedorchuk

created by calamai on 03 Nov 2015

9 nov 2015
-- 14:00

sala conferenze Tricerri, DiMaI, Firenze

**Abstract.**

The famous Mather-Yau theorem says that two isolated
hypersurface singularities are biholomorphically equivalent if and
only if their moduli algebras are isomorphic. However, the
reconstruction problem of explicitly recovering the singularity from
its moduli algebra is open. In the case of quasi-homogeneous
hypersurface singularities, Eastwood and Isaev proposed an
invariant-theoretic approach to the reconstruction problem. For
homogeneous hypersurface singularities, Alper and Isaev gave a
geometric invariant theory reformulation of this approach, in which
GIT stability of the Hilbert points of the Milnor algebra of the
singularity plays a key role. In particular, Alper and Isaev pose
several problems concerning GIT stability of the associated form and
the gradient point of a homogeneous form, which they recently solved
in the binary case and in the case of generic forms in any number of
variables. In my talk, I will explain these recent developments. I
will then proceed to prove semistability of the gradient point of a
semistable form, and semistability of the associated form of a
non-degenerate form. This will answer several (but not all) questions
of Alper and Isaev.