In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, enumerative geometry is the branch of
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
concerned with counting numbers of solutions to geometric questions, mainly by means of
intersection theory
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem o ...
.
History
![Apollonius8ColorMultiplyV2](https://upload.wikimedia.org/wikipedia/commons/8/8c/Apollonius8ColorMultiplyV2.svg)
The
problem of Apollonius
In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (c. 262 190 BC) posed and solved this famous problem in his work (', "Tangencies ...
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
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.
For affine and projective algebraic varieties, the codimension equals ...
*
Bézout's theorem
Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the deg ...
*
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 (part of enumerative geometry). It was a precursor of ...
, and more generally
characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes ...
es in
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
* The connection of counting intersections with cohomology is
Poincaré duality
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...
* The study of
moduli spaces
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...
of curves, maps and other geometric objects, sometimes via the theory of
quantum cohomology In 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, t ...
. The study of
quantum cohomology In 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, t ...
,
Gromov–Witten invariant
In mathematics, specifically in symplectic topology and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations, count pseudoholomorphic curves meeting prescribed conditions in a given symplectic man ...
s and
mirror symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.
In 2D ther ...
gave a significant progress in
Clemens conjecture.
Enumerative geometry is very closely tied to
intersection theory
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem o ...
.
Schubert calculus
Enumerative geometry saw spectacular development towards the end of the nineteenth century, at the hands of
Hermann Schubert
__NOTOC__
Hermann Cäsar Hannibal Schubert (22 May 1848 – 20 July 1911) 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 n ...
. He introduced for the purpose the
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 (part of enumerative geometry). It was a precursor of ...
, which has proved of fundamental geometrical and
topological
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
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 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 (more than 2) curves, and accounting properly for ta ...
s had been rigorously defined (by
André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
as part of his foundational programme 1942–6, and again subsequently), but this did not exhaust the proper domain of enumerative questions.
Fudge factors and Hilbert's fifteenth problem
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 factor A fudge factor is an ''ad hoc'' quantity or element introduced into a calculation, formula or model in order to make it fit observations or expectations. Also known as a "Correction Coefficient" which is defined by:
: \kappa_c = \frac
Examples ...
s", which were only rigorously justified decades later.
As an example, count the
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
s tangent to five given lines in the
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 (namely, parallel lines) that do ...
. The conics constitute a
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
of dimension 5, taking their six coefficients as
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
, and
five points determine a conic
In Euclidean and projective geometry, just as two (distinct) points determine a line (a degree-1 plane curve), five points determine a conic (a degree-2 plane curve). There are additional subtleties for conics that do not exist for lines, and thu ...
, if the points are in
general linear position
In algebraic geometry and computational 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 ...
, 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 mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...
in ''P''
5. However the
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.
These arose first in the f ...
consisting of all such quadrics is not without a
base locus
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.
These arose first in the f ...
. In fact each such quadric contains the
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 Giu ...
, which parametrizes the conics
:(''aX'' + ''bY'' + ''cZ'')
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 = 2
5 points. But the relevant quadrics here are not in
general position
In algebraic geometry and computational 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 ar ...
. 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 A fudge factor is an ''ad hoc'' quantity or element introduced into a calculation, formula or model in order to make it fit observations or expectations. Also known as a "Correction Coefficient" which is defined by:
: \kappa_c = \frac
Examples ...
'.
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. The problem is to put Schubert's enumerative calculus on a rigorous foundation.
Introduction
Schubert calculus is the ...
was to overcome the apparently arbitrary nature of these interventions; this aspect goes beyond the foundational question of the Schubert calculus itself.
Clemens conjecture
In 1984
H. Clemens studied the counting of the number of
rational curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
s on a
quintic threefold In mathematics, a quintic threefold is a 3-dimensional hypersurface of degree 5 in 4-dimensional projective space \mathbb^4. Non-singular quintic threefolds are Calabi–Yau manifolds.
The Hodge diamond of a non-singular quintic 3-fold is
Mathem ...
and reached the following conjecture.
: Let
be a general quintic threefold,
a positive integer, then there are only a finite number of rational curves with degree
on
.
This conjecture has been resolved in the case
, but is still open for higher
.
In 1991 the paper
[* ] about mirror symmetry on the quintic threefold in
from the string theoretical viewpoint gives numbers of degree d rational curves on
for all
. Prior to this, algebraic geometers could calculate these numbers only for
.
Examples
Some of the historically important examples of enumerations in algebraic geometry include:
*2 The number of lines meeting 4 general lines in space
*8 The number of circles tangent to 3 general circles (the
problem of Apollonius
In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (c. 262 190 BC) posed and solved this famous problem in his work (', "Tangencies ...
).
*27 The number of lines on a smooth
cubic surface
In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather th ...
(
Salmon
Salmon () is the common name for several list of commercially important fish species, commercially important species of euryhaline ray-finned fish from the family (biology), family Salmonidae, which are native to tributary, tributaries of the ...
and
Cayley)
*2875 The number of lines on a general
quintic threefold In mathematics, a quintic threefold is a 3-dimensional hypersurface of degree 5 in 4-dimensional projective space \mathbb^4. Non-singular quintic threefolds are Calabi–Yau manifolds.
The Hodge diamond of a non-singular quintic 3-fold is
Mathem ...
*3264 The number of
conics tangent to 5 plane conics in general position (
Chasles)
*609250 The number of conics on a general
quintic threefold In mathematics, a quintic threefold is a 3-dimensional hypersurface of degree 5 in 4-dimensional projective space \mathbb^4. Non-singular quintic threefolds are Calabi–Yau manifolds.
The Hodge diamond of a non-singular quintic 3-fold is
Mathem ...
*4407296 The number of conics tangent to 8 general quadric surfaces
*666841088 The number of quadric surfaces tangent to 9 given quadric surfaces in general position in 3-space
*5819539783680 The number of twisted cubic curves tangent to 12 given quadric surfaces in general position in 3-space
References
*
*
External links
*{{cite journal, author=Bashelor, Andrew, author2=Ksir, Amy, author3=Traves, Will, title=Enumerative Algebraic Geometry of Conics, journal=Amer. Math. Monthly, volume=115, issue=8, year=2008, pages=701–7, url=http://www.maa.org/programs/maa-awards/writing-awards/enumerative-algebraic-geometry-of-conics, jstor=27642583, doi=10.1080/00029890.2008.11920584
Intersection theory
Algebraic geometry