Singular

D.15.24 schubert_lib

Library:

schubert.lib

Purpose:

Proceduces for Intersection Theory

Author:

Hiep Dang, email: hiep@mathematik.uni-kl.de

Overview:

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.

Procedures:

D.15.24.1 makeVariety		create a variety
D.15.24.2 printVariety		print procedure for a variety
D.15.24.3 productVariety		make the product of two varieties
D.15.24.4 ChowRing		create the Chow ring of a variety
D.15.24.5 Grassmannian		create a Grassmannian as a variety
D.15.24.6 projectiveSpace		create a projective space as a variety
D.15.24.7 projectiveBundle		create a projective bundle as a variety
D.15.24.8 integral		degree of a 0-cycle on a variety
D.15.24.9 makeSheaf		create a sheaf
D.15.24.10 printSheaf		print procedure for sheaves
D.15.24.11 rankSheaf		return the rank of a sheaf
D.15.24.12 totalChernClass		compute the total Chern class of a sheaf
D.15.24.13 ChernClass		compute the k-th Chern class of a sheaf
D.15.24.14 topChernClass		compute the top Chern class of a sheaf
D.15.24.15 totalSegreClass		compute the total Segre class of a sheaf
D.15.24.16 dualSheaf		make the dual of a sheaf
D.15.24.17 tensorSheaf		make the tensor of two sheaves
D.15.24.18 symmetricPowerSheaf		make the k-th symmetric power of a sheaf
D.15.24.19 quotSheaf		make the quotient of two sheaves
D.15.24.20 addSheaf		make the direct sum of two sheaves
D.15.24.21 makeGraphVE		create a graph from a list of vertices and a list of edges
D.15.24.22 printGraphG		print procedure for graphs
D.15.24.23 moduliSpace		create a moduli space of stable maps as an algebraic stack
D.15.24.24 printStack		print procedure for stacks
D.15.24.25 dimStack		compute the dimension of a stack
D.15.24.26 fixedPoints		compute the list of graphs corresponding the fixed point components of a torus action on the stack
D.15.24.27 contributionBundle		compute the contribution bundle on a stack at a graph
D.15.24.28 normalBundle		compute the normal bundle on a stack at a graph
D.15.24.29 multipleCover		compute the contribution of multiple covers of a smooth rational curve as a Gromov-Witten invariant
D.15.24.30 linesHypersurface		compute the number of lines on a general hypersurface
D.15.24.31 rationalCurve		compute the Gromov-Witten invariant corresponding the number of rational curves on a general Calabi-Yau threefold
D.15.24.32 sumofquotients		prepare a command for parallel computation
D.15.24.33 homog_part		compute a homogeneous component of a polynomial.
D.15.24.34 homog_parts		compute the sum of homogeneous components of a polynomial
D.15.24.35 logg		compute Chern characters from total Chern classes.
D.15.24.36 expp		compute total Chern classes from Chern characters
D.15.24.37 SchubertClass		compute the Schubert classes on a Grassmannian
D.15.24.38 dualPartition		compute the dual of a partition