COUNTING RATIONAL PLANE CURVES
ISRAEL VAINSENCHER
Departamento de Matemática, CCEN, UFPE, 50740-540 Recife, PE.
This is a repport on work by J. Kock. A rational plane curve of degree d is the image of a nonconstant map t
(x(t), y(t)), where x(t), y(t) are rational functions (quotient of polynomials) with maximal degree d. Let M(0, d ) be the space parametrizing the family of rational plane curves of degree d. This is a variety of dimension 3d - 1. The characteristic number n(d;a, b, c) of such plane curves containing a general points, tangent to b general lines and tangent to c other general lines at marked points is finite provided a + b + 2c = 3d - 1. Until quite recently only the first few were known. Using techniques of stable maps and ideas first developed by M. Kontsevich [Ko], Joachim Kock [JK] obtained in his doctoral thesis WVDD-like PDE's for the tangent potential, e.g.,
G(u1, u2, v1, v2) =
u
/!v
/!d
(1)n(d;2,1,2),
where the sum runs over all d > 0 and nonnegative vectors 
= (1,2), = (1,2). Those PDE's allow for effective recursive computation of the characteristic numbers. (May 19, 2000)
Publication Dates
-
Publication in this collection
05 Jan 2001 -
Date of issue
Dec 2000
