Enumerating stationary multiple-points

Abstract

Given a map f: X -+ Y, an r-fold point off is a point xi of X such that there exist points x2, . . . . x, of X which all have the same image underf: We say that an r-fold point is strict if x, , . . . . x, are all distinct points of X and stationary if some of the xi have coalesced to lie “infinitely near.” The infinitely near points determine tangent directions along the Iibref-‘f(x,), in the sense that they constitute ramification points of f: This paper enlarges on the work of [9] to develop, under appropriate hypotheses, enumerative formulas for loci of arbitrary stationary r-fold points in terms of invariants of the map$ Multiple-point theory has a long and rich history. We may trace work back to Clebsch (1864), who gave a formula for the number of nodes of a plane curve, in terms of the degree and genus of the curve. Perhaps the most famous example of a stationary multiple-point result is the Riemann-Hurwitz formula. Let f: X + Y be a finite, separable, surjective map between smooth projective curves. The Riemann-Hurwitz theorem gives a formula for deg R, where R is the ramification divisor of J: It may be easily seen that deg R is simply a weighted number of stationary doublepoints off: Modern multiple-point theory is concerned with developing formulas for general classes of maps. The subject has been of considerable interest in differential geometry and topology as well as in algebraic geometry. Surveys of the work in these fields can be found in [lo], [9, Introduction], or [8, Chap. V, pp. 365-3911. In algebraic geometry, multiple-point theory has presently culiminated in two basic approaches: the Hilbert scheme (see [lo] for a survey) and the method of iteration. We shall only be concerned with the latter technique in the body of this work. Both approaches share a 149 OOOl-8708/87 $7.50