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D.15.33 schubert_lib

Proceduces for Intersection Theory

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

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.

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.


D.15.33.1 makeVariety  create a variety
D.15.33.2 printVariety  print procedure for a variety
D.15.33.3 productVariety  make the product of two varieties
D.15.33.4 ChowRing  create the Chow ring of a variety
D.15.33.5 Grassmannian  create a Grassmannian as a variety
D.15.33.6 projectiveSpace  create a projective space as a variety
D.15.33.7 projectiveBundle  create a projective bundle as a variety
D.15.33.8 integral  degree of a 0-cycle on a variety
D.15.33.9 makeSheaf  create a sheaf
D.15.33.10 printSheaf  print procedure for sheaves
D.15.33.11 rankSheaf  return the rank of a sheaf
D.15.33.12 totalChernClass  compute the total Chern class of a sheaf
D.15.33.13 ChernClass  compute the k-th Chern class of a sheaf
D.15.33.14 topChernClass  compute the top Chern class of a sheaf
D.15.33.15 totalSegreClass  compute the total Segre class of a sheaf
D.15.33.16 dualSheaf  make the dual of a sheaf
D.15.33.17 tensorSheaf  make the tensor of two sheaves
D.15.33.18 symmetricPowerSheaf  make the k-th symmetric power of a sheaf
D.15.33.19 quotSheaf  make the quotient of two sheaves
D.15.33.20 addSheaf  make the direct sum of two sheaves
D.15.33.21 makeGraphVE  create a graph from a list of vertices and a list of edges
D.15.33.22 printGraphG  print procedure for graphs
D.15.33.23 moduliSpace  create a moduli space of stable maps as an algebraic stack
D.15.33.24 printStack  print procedure for stacks
D.15.33.25 dimStack  compute the dimension of a stack
D.15.33.26 fixedPoints  compute the list of graphs corresponding the fixed point components of a torus action on the stack
D.15.33.27 contributionBundle  compute the contribution bundle on a stack at a graph
D.15.33.28 normalBundle  compute the normal bundle on a stack at a graph
D.15.33.29 multipleCover  compute the contribution of multiple covers of a smooth rational curve as a Gromov-Witten invariant
D.15.33.30 linesHypersurface  compute the number of lines on a general hypersurface
D.15.33.31 rationalCurve  compute the Gromov-Witten invariant corresponding the number of rational curves on a general Calabi-Yau threefold
D.15.33.32 sumofquotients  prepare a command for parallel computation
D.15.33.33 homog_part  compute a homogeneous component of a polynomial.
D.15.33.34 homog_parts  compute the sum of homogeneous components of a polynomial
D.15.33.35 logg  compute Chern characters from total Chern classes.
D.15.33.36 expp  compute total Chern classes from Chern characters
D.15.33.37 SchubertClass  compute the Schubert classes on a Grassmannian
D.15.33.38 dualPartition  compute the dual of a partition