### D.5.17 schubert_lib

Library:
  schubert.lib
Purpose:
  Procedures 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.5.17.1 makeVariety |  | create a variety
> D.5.17.2 printVariety |  | print procedure for a variety
> D.5.17.3 productVariety |  | make the product of two varieties
> D.5.17.4 ChowRing |  | create the Chow ring of a variety
> D.5.17.5 Grassmannian |  | create a Grassmannian as a variety
> D.5.17.6 projectiveSpace |  | create a projective space as a variety
> D.5.17.7 projectiveBundle |  | create a projective bundle as a variety
> D.5.17.8 integral |  | degree of a 0-cycle on a variety
> D.5.17.9 makeSheaf |  | create a sheaf
> D.5.17.10 printSheaf |  | print procedure for sheaves
> D.5.17.11 rankSheaf |  | return the rank of a sheaf
> D.5.17.12 totalChernClass |  | compute the total Chern class of a sheaf
> D.5.17.13 ChernClass |  | compute the k-th Chern class of a sheaf
> D.5.17.14 topChernClass |  | compute the top Chern class of a sheaf
> D.5.17.15 totalSegreClass |  | compute the total Segre class of a sheaf
> D.5.17.16 dualSheaf |  | make the dual of a sheaf
> D.5.17.17 tensorSheaf |  | make the tensor of two sheaves
> D.5.17.18 symmetricPowerSheaf |  | make the k-th symmetric power of a sheaf
> D.5.17.19 quotSheaf |  | make the quotient of two sheaves
> D.5.17.20 addSheaf |  | make the direct sum of two sheaves
> D.5.17.21 makeGraphVE |  | create a graph from a list of vertices and a list of edges
> D.5.17.22 printGraphG |  | print procedure for graphs
> D.5.17.23 moduliSpace |  | create a moduli space of stable maps as an algebraic stack
> D.5.17.24 printStack |  | print procedure for stacks
> D.5.17.25 dimStack |  | compute the dimension of a stack
> D.5.17.26 fixedPoints |  | compute the list of graphs corresponding the fixed point components of a torus action on the stack
> D.5.17.27 contributionBundle |  | compute the contribution bundle on a stack at a graph
> D.5.17.28 normalBundle |  | compute the normal bundle on a stack at a graph
> D.5.17.29 multipleCover |  | compute the contribution of multiple covers of a smooth rational curve as a Gromov-Witten invariant
> D.5.17.30 linesHypersurface |  | compute the number of lines on a general hypersurface
> D.5.17.31 rationalCurve |  | compute the Gromov-Witten invariant corresponding the number of rational curves on a general Calabi-Yau threefold
> D.5.17.32 sumofquotients |  | prepare a command for parallel computation
> D.5.17.33 homog_part |  | compute a homogeneous component of a polynomial.
> D.5.17.34 homog_parts |  | compute the sum of homogeneous components of a polynomial
> D.5.17.35 logg |  | compute Chern characters from total Chern classes.
> D.5.17.36 expp |  | compute total Chern classes from Chern characters
> D.5.17.37 SchubertClass |  | compute the Schubert classes on a Grassmannian
> D.5.17.38 dualPartition |  | compute the dual of a partition

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