Enumerative geometry
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In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory.
History
The problem of Apollonius is one of the earliest examples of enumerative geometry. This problem asks for the number and construction of circles that are tangent to three given circles, points or lines. In general, the problem for three given circles has eight solutions, which can be seen as 23, each tangency condition imposing a quadratic condition on the space of circles. However, for special arrangements of the given circles, the number of solutions may also be any integer from 0 (no solutions) to six; there is no arrangement for which there are seven solutions to Apollonius' problem.
Key tools
A number of tools, ranging from the elementary to the more advanced, include:
- Dimension counting
- Bézout's theorem
- Schubert calculus, and more generally characteristic classes in cohomology
- The connection of counting intersections with cohomology is Poincaré duality
- The study of Clemens conjecture.
Enumerative geometry is very closely tied to intersection theory.
Schubert calculus
Enumerative geometry saw spectacular development towards the end of the nineteenth century, at the hands of
Fudge factors and Hilbert's fifteenth problem
Naïve application of dimension counting and Bézout's theorem yields incorrect results, as the following example shows. In response to these problems, algebraic geometers introduced vague "fudge factors", which were only rigorously justified decades later.
As an example, count the
- (aX + bY + cZ)2 = 0
called 'double lines'. This is because a double line intersects every line in the plane, since lines in the projective plane intersect, with multiplicity two because it is doubled, and thus satisfies the same intersection condition (intersection of multiplicity two) as a nondegenerate conic that is tangent to the line.
The general
Hilbert's fifteenth problem was to overcome the apparently arbitrary nature of these interventions; this aspect goes beyond the foundational question of the Schubert calculus itself.
Clemens conjecture
In 1984
- Let be a general quintic threefold, a positive integer, then there are only a finite number of rational curves with degree on .
This conjecture has been resolved in the case , but is still open for higher .
In 1991 the paper[4] about mirror symmetry on the quintic threefold in from the string theoretical viewpoint gives numbers of degree d rational curves on for all . Prior to this, algebraic geometers could calculate these numbers only for .
Examples
Some of the historically important examples of enumerations in algebraic geometry include:
- 2 The number of lines meeting 4 general lines in space
- 8 The number of circles tangent to 3 general circles (the problem of Apollonius).
- 27 The number of lines on a smooth cubic surface (Salmon and Cayley)
- 2875 The number of lines on a general quintic threefold
- 3264 The number of conics tangent to 5 plane conics in general position (Chasles)
- 609250 The number of conics on a general quintic threefold
- 4407296 The number of conics tangent to 8 general quadric surfaces Fulton (1984, p. 193)
- 666841088 The number of quadric surfaces tangent to 9 given quadric surfaces in general position in 3-space (Schubert 1879, p.106) (Fulton 1984, p. 193)
- 5819539783680 The number of twisted cubic curves tangent to 12 given quadric surfaces in general position in 3-space (Schubert 1879, p.184) (S. Kleiman, S. A. Strømme & S. Xambó 1987)
References
- ^ Schubert, H. (1879). Kalkül der abzählenden Geometrie (published 1979).
- ISBN 9780821874622.
- ISBN 0-387-12176-5.
- .
- Kleiman, S.; Strømme, S. A.; Xambó, S. (1987), "Sketch of a verification of Schubert's number 5819539783680 of twisted cubics", Space curves (Rocca di Papa, 1985), Lecture Notes in Math., vol. 1266, Berlin: Springer, pp. 156–180, MR 0908713
- Schubert, Hermann (1979) [1879], Kleiman, Steven L. (ed.), Kalkül der abzählenden Geometrie, Reprint of the 1879 original (in German), Berlin-New York: Springer-Verlag, MR 0555576
External links
- Bashelor, Andrew; Ksir, Amy; Traves, Will (2008). "Enumerative Algebraic Geometry of Conics". Amer. Math. Monthly. 115 (8): 701–7. JSTOR 27642583.