Math.RootLoci.Dual.Restriction

Formula for the dual cohomology class of the cones over the strata (sometimes called Thom polynomial) in terms of the Chern classes c1 and c2, from the author's MSc thesis.

Note that the dual class agress with the lowest degree part of the CSM class.

See: Balazs Komuves: Thom Polynomials via Restriction Equations; MSc thesis, ELTE, 2003

The dual class

affineDualMSc :: Partition -> ZMod Chern

The affine Thom polynomial formula from my MSc thesis

Degree

projDegreeFromDual

Arguments

:: Int

number of points = dimension of the projective space P^n

-> ZMod Chern

dual class

-> Integer

degree

Compute the projective degree from the affine equivariant dual (which can be checked against Hilbert's formula)

This is just a simple substition:

alpha  ->  1/n
beta   ->  1/n

or in terms of Chern classes:

c1     ->  2/n
c2     ->  1/n^2

degreeMSc :: Partition -> Integer

Compute the degree of the strata via the formula for the dual class

extract the dual class from the CSM class

dualClassFromProjCSM :: forall base. ChernBase base => ZMod (Gam base) -> ZMod base

The dual class of the closure agress with the lowest degree part of the CSM class.

dualClassFromAffCSM :: ChernBase base => ZMod base -> ZMod base

Lemma 9.1.3

lemma913 :: Partition -> Int -> Bool

Checks if Lemma 9.1.3 from the thesis is true for the given inputs

helper functions

divideIntoTwo :: Partition -> Set (Partition, Partition)

Different ways to divide a partition into two

divideIntoTwoNonEmpty :: Partition -> Set (Partition, Partition)

Different ways to divide a partition into two nonempty partitions