A Construction of Multiplicity Class of Hypersurfaces From Hesselink Stratification of a Hilbert Scheme

ABSTRACT

It is well known that there is a positive relationship between the maximal multiplicity and the length of associated virtual 1-parameter subgroup of a projective hypersurface. In this article, we will define the multiplicity classes of hypersurfaces and construct them from the Hesselink stratification of a Hilbert scheme.

KEYWORDS:

• Hesselink stratification
• Hypersurface singularity
• Geometric invariant theory

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 14L24

Notes

1 According to the literature, such a definition of state polytope is obtained from the canonical ${\text{GL}}_{r+1}(k)$-action with the canonical linearization twisted by a power of determinant while we are considering the canonical ${\text{SL}}_{r+1}(k)$-action. However, such a setup does not changes the situation of our problem in the viewpoint of GIT as we can see in [4, 2.2]. Also, such a definition of state polytope let us observe the symmetry within our problem directly from the picture.

Additional information

Funding

This work had been supported by a KIAS individual grant (6G067904) at Korea Institute for Advanced Study.