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D.5.17 schubert_lib
- Library:
- schubert.lib
- Purpose:
- Procedures for Intersection Theory
- Author:
- Hiep Dang, email: [email protected]
- 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