Enumerative geometry

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.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.A number of tools, ranging from the elementary to the more advanced, include:Enumerative geometry saw spectacular development towards the end of the nineteenth century, at the hands of Hermann Schubert. He introduced for the purpose the Schubert calculus, which has proved of fundamental geometrical and topological value in broader areas. The specific needs of enumerative geometry were not addressed until some further attention was paid to them in the 1960s and 1970s (as pointed out for example by Steven Kleiman). Intersection numbers had been rigorously defined (by André Weil as part of his foundational programme 1942–6, and again subsequently), but this did not exhaust the proper domain of enumerative questions.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.In 1984 H. Clemens studied the counting of the number of rational curves on a quintic threefold X ⊂ P 4 {displaystyle Xsubset P^{4}}   and reached the following conjecture.Some of the historically important examples of enumerations in algebraic geometry include: