We compute the number of irreducible rational curves of given degree with 1 tacnode in or 1 node in meeting an appropriate generic collection of points and lines. As a byproduct, we also compute the number of rational plane curves of degree passing through given points and tangent to a given line. The method is ‘classical’, free of Quantum Cohomology.
@article{ASNSP_2004_5_3_1_67_0, author = {Ran, Ziv}, title = {Enumerative geometry of divisorial families of rational curves}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {67--85}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 3}, number = {1}, year = {2004}, mrnumber = {2064968}, zbl = {1170.14306}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2004_5_3_1_67_0/} }
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 SP - 67 EP - 85 VL - 3 IS - 1 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2004_5_3_1_67_0/ LA - en ID - ASNSP_2004_5_3_1_67_0 ER -
%0 Journal Article %A Ran, Ziv %T Enumerative geometry of divisorial families of rational curves %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2004 %P 67-85 %V 3 %N 1 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2004_5_3_1_67_0/ %G en %F ASNSP_2004_5_3_1_67_0
Ran, Ziv. Enumerative geometry of divisorial families of rational curves. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 3 (2004) no. 1, pp. 67-85. http://archive.numdam.org/item/ASNSP_2004_5_3_1_67_0/
[CR] Dimension of families of space curves, Comp. Math. 90 (1994), 53-57. | Numdam | MR | Zbl
- ,[DH] Geometry of Severi varieties, Trans. Amer. Math. Soc. 309 (1988) 1-34. | MR | Zbl
- ,[F] “Intersection theory”, Springer, 1984. | MR | Zbl
,[FP] 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. | MR | Zbl
- ,[KP] Enumerating singular curves on surfaces, In: “Algebraic geometry”, Hirzebruch 70, Contemporary Math. 241 (1999), 209-238, corrections and revision in math.AG/9903192. | MR | Zbl
- ,[P] The canonical bundle of and enumerative geometry, Internat. Math. Res. Notices, 1997, 173-186. | MR | Zbl
,[R1] Enumerative geometry of singular plane curves, Invent. Math. 97 (1987), 447-465. | MR | Zbl
,[R2] Bend, break and count, Israel J. Math. 111 (1999), 109-124. | MR | Zbl
,[R3] Bend, break and count II, Math. Proc. Camb. Phil. Soc. 127 (1999), 7-12. | MR | Zbl
,[R4] On the variety of rational space curve, Israel J. Math. 122 (2001), 359-370. | MR | Zbl
,[R5] The degree of the divisor of jumping rational curves, Quart. J. Math. (2001), 1-18. | MR | Zbl
,[R6] Geometry on nodal curves, (preprint math.AG/0210209). | MR
,[Z] Almindelige Egenskaber ved Systemer af plane Kurver, Vidensk. Selsk. Skr., 5 Række, naturvidenskabelig og mathematisk Afd., Nr. 10, B. IV, København (1873), 286-393. | JFM
,[Zi] Enumeration of 1-nodal rational curves in projective spaces, math.AG 0204236.
,