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.

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, 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 by Steven Kleiman). Intersection numbers had been rigorously defined (by André Weil as part of his foundational programme 1942–6, and again subsequently). This did not exhaust the proper domain of enumerative questions.

William Fulton gives the following example: count the conic sections tangent to five given lines in the projective plane. The conics constitute a projective space of dimension 5, taking their six coefficients as homogeneous coordinates. Tangency to a given line L is one condition, so determined a quadric in P5. However the linear system of divisors consisting of all such quadrics is not without a base locus. In fact each such quadric contains the Veronese surface, which parametrizes the conics

(aX + bY + cZ)² = 0

called 'double lines'. The general Bézout theorem says 5 quadrics will intersect in 32 = 2⁵ points. But the relevant quadrics here are not in general 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.

Enumerative Geometry - Barbara Fantechi - 2015

References

* H. Schubert, Kalkul der abzählenden Geometrie (1879) reprinted 1979.

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