Automating the construction of stationary multiple-point classes

ABSTRACT

In this paper, we describe an algorithm to compute arbitrary stationary multiple-point formulas. We report its full implementation in Maple and show some examples matching formulas found by hand computation. We also present an application to the enumeration of lines having specified contact with a projective surface.

REFERENCES

• 1: S.J. Colley. On the Enumerative Geometry o} Stationary Multiple-Points. PhD thesis, Massachusetts Institute of Technology, May 1983.
• 2: S.3. Colley. Enumerating Stationary Multiple-Points. Advances in Mathematics, 66(2):149-170, 1987.
• 3: W. Fulton. Intersection Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, 1984.
• 4: P. Griffiths and J. Harris. Princwles o} Algebraic Geometry. 3. Wiley and Sons, New York, 1978.
• 5: S. Katz. iteration of Multiple-Point Formulas and Appfications to Conics. In Proceedings Sundance, volume 1311 of Lecture Notes in Mathematics, pages 147-155. Springer-Voting, 1986.
• 6: S.L. Kleiman. Multiple-Point Formulas I" Iteration. Acta Mathematica, 147:13-49, 1981.
• 7: S.L. Kleiman. Multiple-Point Formulas II: The Hilbert Scheme. in S. Xamb&Descamps, editor, Enumerative Geometry, volume 1436 of Lecture Notes in Mathematics, pages 101-138. Springer-Verlag, 1990. Proceedings, Sitges 1987.
• 8: P. le Barz. Formules pour los tris6cantes des surfaces alg6briques. L 'Enseignement Mathdmatique, 33"1-66, 1987.
• 9: MAPLE V. Copyright (c) 1981-1990 by the University of Waterloo. All rights reserved. MAPLE is a registered trademark of Waterloo Maple Software.
• 10: S. Petitjean. The Complexity and Enumerative Geometry of Aspect Graphs of Smooth Surfaces. In Proceedings of MEGA '95, Santander, Spain, 1994. To appear.
• 11: G. Salmon. A Treatise on the Analytic Geometry of Three Dimensions, volume iI. Dublin, 5th edition, 1915.

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Algebraic algorithms

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Geometrical problems and computations

  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.3 Languages and Systems

          Nouns: Maple

General Terms:
Algorithms, Languages, Theory