- Enumerative geometry
In

mathematics ,**enumerative geometry**is the branch ofalgebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means ofintersection 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 2^{3}, 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 generallycharacteristic class es incohomology

* The connection of counting intersections with cohomology isPoincaré duality

* The study ofmoduli spaces of curves, maps and other geometric objects, sometimes via the theory ofquantum cohomology .Enumerative geometry is very closely tied to intersection theory.

**chubert calculus**Enumerative geometry saw spectacular development towards the end of the

nineteenth century , at the hands ofHermann Schubert . He introduced for the purpose theSchubert calculus , which has proved of fundamental geometrical andtopological value in broader areas. The specific needs of enumerative geometry were not addressed, in the general assumption that algebraic geometry had been fully axiomatised, until some further attention was paid to them in the 1960s and 1970s (as pointed out for example bySteven Kleiman ).Intersection number s had been rigorously defined (byAndré Weil as part of his foundational programme 1942–6, and again subsequently). This did not exhaust the proper domain of enumerative questions.**Fudge factors and Hilbert's fifteenth problem**Naïve application of dimension counting and Bezout’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.

William Fulton gives the following example: count the

conic section s tangent to five given lines in theprojective plane . The conics constitute aprojective space of dimension 5, taking their six coefficients ashomogeneous coordinates . Tangency to a given line "L" is one condition, so determined aquadric in "P"^{5}. However thelinear system of divisors consisting of all such quadrics is not without abase locus . In fact each such quadric contains theVeronese surface , which parametrizes the conics:("aX" + "bY" + "cZ")

^{2}= 0called 'double lines'. The general

Bézout theorem says 5 quadrics will intersect in 32 = 2^{5}points. But the relevant quadrics here are not ingeneral position . From 32, 31 must be subtracted and attributed to the Veronese, to leave the correct answer (from the point of view of geometry), namely 1. This process of attributing intersections to 'degenerate' cases is a typical geometric introduction of a 'fudge factor '.It was a

Hilbert problem (the fifteenth, in a more stringent reading) to overcome the apparently arbitrary nature of these interventions; this aspect goes beyond the foundational question of the Schubert calculus itself.**References***H. Schubert, "Kalkul der abzählenden Geometrie" (1879) reprinted 1979.

*William Fulton, "Intersection Theory" (1984), Chapter 10.4

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