Math.RootLoci.CSM.Aluffi

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).

CSM computation

aluffiOpenCSM :: Partition -> ZMod G

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

aluffiClosedCSM :: Partition -> ZMod G

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

Euler characteristics

aluffiOpenEuler :: Partition -> Integer

Euler characteristic, computed form aluffiOpenCSM

aluffiClosedEuler :: Partition -> Integer

Euler characteristic, computed form aluffiClosedCSM

openEulerChar :: Partition -> Integer

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

csmToEulerOfLinearSections

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

aluffiDegree :: Partition -> Integer

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