## Some libraries for Singular

**chern.lib**: a library for symbolic computations with Chern classes.### 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.

The first 38 terms of the Chern character.

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

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.

**lrcalc.lib**: a Singular interface to the Littlewood-Richardson Calculator by Anders Buch.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.-
**goettsche.lib**: an implementation of the Drezét's, Göttsche's, and Macdonald's formulas.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.