# Multiple Point Formulas for Maps

• Steven L. Kleiman
Chapter
Part of the Progress in Mathematics book series (PM, volume 24)

## Abstract

Let f: X → Y be a map. Its set of r-fold points is
$$\begin{array}{*{20}{c}} {{{\text{M}}_{{\text{r}}}}{\text{ = }}\{ {\text{x}}\varepsilon {\text{X|}}} & {{\text{there}} {\text{exist}}} & {{{\text{x}}_{2}},...,{{\text{x}}_{{\text{r}}}}} & {{\text{with}}} & {{\text{f(}}{{\text{x}}_{{\text{i}}}}){\text{ = f}}({\text{x}})\} ;} \\ \end{array}$$
the xi must be distinct from x and from each other, but they may lie “infinitely close” (that is, determine tangent directions along the fiber f-1f(x)). An r-fold-point formula is a polynomial expression in the invariants of f which gives, under appropriate hypotheses, the number of r-fold points, weighted by natural multiplicities, or the class mr of a natural positive cycle supported by Mr. The theory of these formulas will be surveyed here, concentrating on some of the author’s recent work, Kleiman [1981b], [1982]. Aside from a few comments, the setting will be algebraic geometry, although the formulas and their proofs have a universal character.

## Keywords

Algebraic Geometry Double Point Intersection Formula Hilbert Scheme Refined Version
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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