## Some libraries for Singular

**chern.lib**: a library for symbolic computations with Chern classes.**lrcalc.lib**: a Singular interface to the Littlewood-Richardson Calculator by Anders Buch.-
**goettsche.lib**: an implementation of the Drezét's, Göttsche's, and Macdonald's formulas.

### Overview

This library is a toolbox for symbolic computations with Chern classes.The Aluffi's algorithms for computation of characteristic classes of algebraic varieties (Segre, Fulton, Chern-Schwartz-MacPherson classes) are implemented as well (cf. doi:10.1016/S0747-7171(02)00089-5).

The current version of the library chern.lib is available on GitHub.

An informal discussion of the algorithms implemented in the library: ps-file , pdf-file .

### On different approaches to compute the Chern classes of a tensor product of two vector bundles

A comparison of different algorithms implemented in the library to compute the Chern classes of a tensor product of two vector bundles: ps-file , pdf-file .Times needed to obtain all the formulas for the Chern classes of tensor products E⊗F with rank(E⊗F)=N.

The Singular code used for testing the four different implementations.

### Some examples

The total Chern class and the Hilbert polynomial of the Horrocks-Mumford bundle (cf. doi:10.1016/0040-9383(73)90022-0).####
Things that can be computed with the help of the library in around 1 second.

(Laptop 2 x Intel Celeron 2957U @ 1.4 GHz with 8.3 GB of RAM)

The
first 38 terms of the Chern character.
The first 16 terms of the Todd class.

The first 12 Chern classes of a tensor product of two bundles.

The Chern classes of all exterior powers of a rank 11 vector bundle of a 3-fold.

The Chern classes of 9 first symmetric powers of a rank 5 vector bundle of a 4-fold.

This library is a Singular interface for
the Littlewood-Richardson Calculator by Anders Buch.
The current version of the library lrcalc.lib is available on
GitHub.

This is an implementation of the following formulas.
The Drezet's formula for the Betti numbers of moduli spaces of Kronecker modules
(cf. doi: 10.1007/BF01449215).

The Göttsche's formula for the Betti numbers of Hilbert schemes of points on a surface (cf. doi: 10.1007/BF01453572).

The Nakajima's and Yoshioka's formula for the Betti numbers of the punctual Quot-schemes on a plane or, equivalently, of the moduli spaces of the framed torsion-free planar sheaves (cf. doi: 10.1090/crmp/038/02).

The Macdonald's formula for the Betti numbers of symmetric products (cf. doi: 10.1017/S0305004100040573). The current version of the library goettsche.lib is available on GitHub.

The Göttsche's formula for the Betti numbers of Hilbert schemes of points on a surface (cf. doi: 10.1007/BF01453572).

The Nakajima's and Yoshioka's formula for the Betti numbers of the punctual Quot-schemes on a plane or, equivalently, of the moduli spaces of the framed torsion-free planar sheaves (cf. doi: 10.1090/crmp/038/02).

The Macdonald's formula for the Betti numbers of symmetric products (cf. doi: 10.1017/S0305004100040573). The current version of the library goettsche.lib is available on GitHub.