Safe Haskell | None |
---|---|

Language | Haskell2010 |

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).

- aluffiOpenCSM :: Partition -> ZMod G
- aluffiClosedCSM :: Partition -> ZMod G
- aluffiOpenEuler :: Partition -> Integer
- aluffiClosedEuler :: Partition -> Integer
- openEulerChar :: Partition -> Integer
- csmToEulerOfLinearSections :: Int -> ZMod G -> [Integer]
- aluffiDegree :: Partition -> Integer

# CSM computation

aluffiOpenCSM :: Partition -> ZMod G Source #

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

aluffiClosedCSM :: Partition -> ZMod G Source #

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

# Euler characteristics

aluffiOpenEuler :: Partition -> Integer Source #

Euler characteristic, computed form `aluffiOpenCSM`

aluffiClosedEuler :: Partition -> Integer Source #

Euler characteristic, computed form `aluffiClosedCSM`

openEulerChar :: Partition -> Integer Source #

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 Source #

:: Int | the dimension of the ambient projective space |

-> 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 Source #

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

generic hiperplanes.