coincident-root-loci-0.2: Equivariant CSM classes of coincident root loci

Math.RootLoci.CSM.Aluffi

Description

Aluffi's computation of the non-equivariant CSM in P^n

See: Paolo Aluffi: Characteristic classes of discriminants and enumerative geometry, Comm. in Algebra 26(10), 3165-3193 (1998).

Synopsis

# CSM computation

Paolo Aluffi's explicit formula for the (non-equivariant) CSM of open coincident root loci

Summing together the open loci CSMs, we got the CSMs of the closures of the strata

# Euler characteristics

Euler characteristic, computed form aluffiOpenCSM

Euler characteristic, computed form aluffiClosedCSM

It is easy to see from Aluffi's formula that only dimensions 1 and 2 has nonzero Euler characteristic. This function implements the resulting rather trivial formula:

chi( X_{n}   ) = 2
chi( X_{p,q} ) = if p==q then 1 else 2
chi( X_{...} ) = 0

# General linear sections

Arguments

 :: Int the dimension of the ambient projective space P^n -> ZMod G the CSM class -> [Integer] the resulting sequence of Euler characteristics

Converts the CSM class of a (locally closed?) projective variety Z to the Euler characteristics of general linear sections of Z (so the first number will be chi(Z), the second will be chi(Z cap H1), the third chi(Z cap H1 cap H2) with H1, H2... being generic hyperplanes. Finally the codim-th number will be the degree.

See: Paolo Aluffi: EULER CHARACTERISTICS OF GENERAL LINEAR SECTIONS AND POLYNOMIAL CHERN CLASSES, Proposition 2.6

We can compute the degree of the closures of the strata by intersection them with dim(X) generic hiperplanes.