Enumerative geometry
Encyclopedia
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.
Enumerative geometry is very closely tied to intersection theory.
. 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 number
s 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.
As an example, count the conic section
s 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
, and five points determine a conic
, if the points are in general linear position, as passing through a given point imposes a linear condition. Similarly, tangency to a given line L (tangency is intersection with multiplicity two) is one quadratic 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
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 Bézout theorem says 5 general quadrics in 5-space 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, enumerative geometry is the branch of algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory
Intersection theory
In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is older, with roots in Bézout's theorem on curves and...
.
History
Problem of Apollonius
In Euclidean plane geometry, Apollonius' problem is to construct circles that are tangent to three given circles in a plane . Apollonius of Perga posed and solved this famous problem in his work ; this work has been lost, but a 4th-century report of his results by Pappus of Alexandria has survived...
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 theoremBézout's theoremBézout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves. The theorem claims that the number of common points of two such curves X and Y is equal to the product of their degrees...
- Schubert calculusSchubert calculusIn mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry...
, and more generally characteristic classCharacteristic classIn mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not...
es in cohomologyCohomologyIn mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries... - The connection of counting intersections with cohomology is Poincaré dualityPoincaré dualityIn mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...
- The study of moduli spaces of curves, maps and other geometric objects, sometimes via the theory of quantum cohomologyQuantum cohomologyIn mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold. It comes in two versions, called small and big; in general, the latter is more complicated and contains more...
.
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 Hermann SchubertHermann Schubert
Hermann Cäsar Hannibal Schubert was a German mathematician.Schubert was one of the leading developers of enumerative geometry, which considers those parts of algebraic geometry that involve a finite number of solutions. In 1874, Schubert won a prize for solving a question posed by Zeuthen...
. He introduced for the purpose the Schubert calculus
Schubert calculus
In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry...
, 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
Steven Kleiman
Steven Lawrence Kleiman is an American mathematician.He is a Professor of Mathematics at the Massachusetts Institute of Technology. Born in Boston, he did his undergraduate studies at the MIT. He received his Ph.D. from Harvard University in 1965, after studying there with Oscar Zariski and David...
). Intersection number
Intersection number
In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple curves, and accounting properly for tangency...
s had been rigorously defined (by André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
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.As an example, count the conic section
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
s tangent to five given lines in the projective plane
Projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
. The conics constitute a projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
of dimension 5, taking their six coefficients as homogeneous coordinates
Homogeneous coordinates
In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
, and five points determine a conic
Five points determine a conic
In geometry, just as two points determine a line , five points determine a conic . There are additional subtleties for conics that do not exist for lines, and thus the statement and its proof for conics are both more technical than for lines.Formally, given any five points in the plane in general...
, if the points are in general linear position, as passing through a given point imposes a linear condition. Similarly, tangency to a given line L (tangency is intersection with multiplicity two) is one quadratic condition, so determined a quadric
Quadric
In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...
in P5. However the linear system of divisors
Linear system of divisors
In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family....
consisting of all such quadrics is not without a base locus
Base locus
In mathematics, specifically algebraic geometry, the base locus of a linear system of divisors on a variety refers to the subvariety of points 'common' to all divisors in the linear system....
. In fact each such quadric contains the Veronese surface
Veronese surface
In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complete linear system of conics. It is named after Giuseppe Veronese...
, which parametrizes the conics
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 Bézout theorem says 5 general quadrics in 5-space will intersect in 32 = 25 points. But the relevant quadrics here are not in general position
General position
In algebraic geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general case situation, as opposed to some more special or coincidental cases that are possible...
. 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
Fudge factor
A fudge factor is a quantity introduced into a calculation in order to "fudge" the results: that is, either to make them match better what happens in the real world, or to add an error margin...
'.
It was a Hilbert problem (the fifteenth
Hilbert's fifteenth problem
Hilbert's fifteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It entails a rigorous foundation of Schubert's enumerative calculus....
, 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.