Enumerative geometry

A number of tools, ranging from the elementary to the more advanced, include: Enumerative geometry is very closely tied to intersection theory.

[1] Enumerative geometry saw spectacular development towards the end of the nineteenth century, at the hands of Hermann Schubert.

[2] He introduced it for the purpose of 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,[3] 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.

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.

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.