We implement new classes (variety, sheaf, stack, graph) and methods for
computing with them. An abstract variety is represented by a nonnegative
integer which is its dimension and a graded ring which is its Chow ring.
An abstract sheaf is represented by a variety and a polynomial which is its
Chern character. In particular, we implement the concrete varieties such as
projective spaces, Grassmannians, and projective bundles.
An important task of this library is related to the computation of
Gromov-Witten invariants. In particular, we implement new tools for the
computation in equivariant intersection theory. These tools are based on the
localization of moduli spaces of stable maps and Bott's formula. They are
useful for the computation of Gromov-Witten invariants. In order to do this,
we have to deal with moduli spaces of stable maps, which were introduced by
Kontsevich, and the graphs corresponding to the fixed point components of a
torus action on the moduli spaces of stable maps.
As an insightful example, the numbers of rational curves on general complete
intersection Calabi-Yau threefolds in projective spaces are computed up to
degree 6. The results are all in agreement with predictions made from mirror
symmetry computations.
References:
Hiep Dang, Intersection theory with applications to the computation of
Gromov-Witten invariants, Ph.D thesis, TU Kaiserslautern, 2013.
Sheldon Katz and Stein A. Stromme, Schubert-A Maple package for intersection
theory and enumerative geometry, 1992.
Daniel R. Grayson, Michael E. Stillman, Stein A. Stromme, David Eisenbud and
Charley Crissman, Schubert2-A Macaulay2 package for computation in
intersection theory, 2010.
Maxim Kontsevich, Enumeration of rational curves via torus actions, 1995.