Enumerative geometry of divisorial families of rational curves

Ziv Ran[1]

  • [1] Mathematics Department University of California Riverside CA 92521, USA

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2004)

  • Volume: 3, Issue: 1, page 67-85
  • ISSN: 0391-173X

Abstract

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We compute the number of irreducible rational curves of given degree with 1 tacnode in 2 or 1 node in 3 meeting an appropriate generic collection of points and lines. As a byproduct, we also compute the number of rational plane curves of degree d passing through 3 d - 2 given points and tangent to a given line. The method is ‘classical’, free of Quantum Cohomology.

How to cite

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Ran, Ziv. "Enumerative geometry of divisorial families of rational curves." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 3.1 (2004): 67-85. <http://eudml.org/doc/84529>.

@article{Ran2004,
abstract = {We compute the number of irreducible rational curves of given degree with 1 tacnode in $¶^2$ or 1 node in $¶^3$ meeting an appropriate generic collection of points and lines. As a byproduct, we also compute the number of rational plane curves of degree $d$ passing through $3d-2$ given points and tangent to a given line. The method is ‘classical’, free of Quantum Cohomology.},
affiliation = {Mathematics Department University of California Riverside CA 92521, USA},
author = {Ran, Ziv},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {1},
pages = {67-85},
publisher = {Scuola Normale Superiore, Pisa},
title = {Enumerative geometry of divisorial families of rational curves},
url = {http://eudml.org/doc/84529},
volume = {3},
year = {2004},
}

TY - JOUR
AU - Ran, Ziv
TI - Enumerative geometry of divisorial families of rational curves
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2004
PB - Scuola Normale Superiore, Pisa
VL - 3
IS - 1
SP - 67
EP - 85
AB - We compute the number of irreducible rational curves of given degree with 1 tacnode in $¶^2$ or 1 node in $¶^3$ meeting an appropriate generic collection of points and lines. As a byproduct, we also compute the number of rational plane curves of degree $d$ passing through $3d-2$ given points and tangent to a given line. The method is ‘classical’, free of Quantum Cohomology.
LA - eng
UR - http://eudml.org/doc/84529
ER -

References

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  1. [CR] M. C. Chang – Z. Ran, Dimension of families of space curves, Comp. Math. 90 (1994), 53-57. Zbl0803.14011MR1266494
  2. [DH] S. Diaz – J. Harris, Geometry of Severi varieties, Trans. Amer. Math. Soc. 309 (1988) 1-34. Zbl0677.14003MR957060
  3. [F] W. Fulton, “Intersection theory”, Springer, 1984. Zbl0541.14005MR732620
  4. [FP] W. Fulton – R. Pandharipande, Notes on stable maps and quantum cohomology, In: “Algebraic Geometry”, Proceedings Santa Cruz 1995, D. R. Morrison (ed.), pp. 45-96. Providence, R.I.: Amer. Math. Soc., 1996. Zbl0898.14018MR1492534
  5. [KP] S. Kleiman – R. Piene, Enumerating singular curves on surfaces, In: “Algebraic geometry”, Hirzebruch 70, Contemporary Math. 241 (1999), 209-238, corrections and revision in math.AG/9903192. Zbl0953.14031MR1718146
  6. [P] R. Pandharipande, The canonical bundle of M ¯ d , 0 and enumerative geometry, Internat. Math. Res. Notices, 1997, 173-186. Zbl0898.14010MR1436774
  7. [R1] Z. Ran, Enumerative geometry of singular plane curves, Invent. Math. 97 (1987), 447-465. Zbl0702.14040MR1005002
  8. [R2] Z. Ran, Bend, break and count, Israel J. Math. 111 (1999), 109-124. Zbl0958.14038MR1710734
  9. [R3] Z. Ran, Bend, break and count II, Math. Proc. Camb. Phil. Soc. 127 (1999), 7-12. Zbl0972.14040MR1692527
  10. [R4] Z. Ran, On the variety of rational space curve, Israel J. Math. 122 (2001), 359-370. Zbl1076.14539MR1826508
  11. [R5] Z. Ran, The degree of the divisor of jumping rational curves, Quart. J. Math. (2001), 1-18. Zbl1076.14530MR1865907
  12. [R6] Z. Ran, Geometry on nodal curves, (preprint math.AG/0210209). MR2157135
  13. [Z] H. Zeuthen, Almindelige Egenskaber ved Systemer af plane Kurver, Vidensk. Selsk. Skr., 5 Række, naturvidenskabelig og mathematisk Afd., Nr. 10, B. IV, København (1873), 286-393. JFM05.0327.01
  14. [Zi] A. Zinger, Enumeration of 1-nodal rational curves in projective spaces, math.AG 0204236. 

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