Susan Jane Colley

Enumerating stationary multiple-points

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

Calculus in the Brewery

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

What is Mathematics and Why Won't it Go Away?

7.50

Calculus, pi, and the Machine Age

(1994). Calculus in the Brewery. The College Mathematics Journal: Vol. 25, No. 3, pp. 226-227.

Cartan Prolongation of a Family of Curves Acquiring a Node

Abstract We report on a seminar for first-year college students that weaves mathematical proof and problem-solving together with discussions of cultural, philosophical, and aesthetic issues surrounding mathematics.

Tangential Quantum Cohomology of Arbitrary Order

Susan Colley (sjcolley@math.oberlin.edu) received her S.B. (1979) and Ph.D. (1983) degrees in mathematics from MIT. Since completing her formal education, she has been a faculty member at Oberlin College. Her research specialty is algebraic geometry, particularly enumerative problems. Besides enjoying time spent with her family and two feline supervisors, her hobbies include old movies, corny music, and attending college committee meetings.

Stationary points of plane forms

Using the monster/Semple tower construction, we study the structure of the Cartan prolongation of the family

The enumeration of simultaneous higher-order contacts between plane curves

x_1x_2 = t

Computing Severi degrees with long-edge graphs

of plane curves with nodal central member.

Coincidence formulas for line complexes

Kock has previously defined a tangency quantum product on formal power series with coefficients in the cohomology ring of any smooth projective variety, and thus a ring that generalizes the quantum cohomology ring. We further generalize Kocks construction by defining a dth-order contact product and establishing its associativity.