In
algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a
scheme or an
algebraic stack) whose points represent isomorphism classes of
algebraic 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 ...
s. It is thus a special case of a
moduli space. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. One also distinguishes between
fine and
coarse moduli spaces for the same moduli problem.
The most basic problem is that of moduli of
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
complete curves of a fixed
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
. Over the
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of
complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
these correspond precisely to
compact Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versio ...
s of the given genus, for which
Bernhard Riemann proved the first results about moduli spaces, in particular their dimensions ("number of parameters on which the complex structure depends").
Moduli stacks of stable curves
The moduli stack
classifies families of smooth projective curves, together with their isomorphisms. When
, this stack may be compactified by adding new "boundary" points which correspond to stable nodal curves (together with their isomorphisms). A curve is
stable if it is complete, connected, has no singularities other than double points, and has only a finite group of automorphisms. The resulting stack is denoted
. Both moduli stacks carry universal families of curves.
Both stacks above have dimension
; hence a stable nodal curve can be completely specified by choosing the values of
parameters, when
. In lower genus, one must account for the presence of smooth families of automorphisms, by subtracting their number. There is exactly one complex curve of genus zero, the Riemann sphere, and its group of isomorphisms is PGL(2). Hence the dimension of
is equal to
:
Likewise, in genus 1, there is a one-dimensional space of curves, but every such curve has a one-dimensional group of automorphisms. Hence, the stack
has dimension 0.
Construction and irreducibility
It is a non-trivial theorem, proved by
Pierre Deligne
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord P ...
and
David Mumford
David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded t ...
,
that the moduli stack
is irreducible, meaning it cannot be expressed as the union of two proper substacks. They prove this by analyzing the locus
of
stable curves in the
Hilbert scheme
In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is ...
:
of tri-canonically embedded curves (from the embedding of the very ample
for every curve) which have
Hilbert polynomial (note: this can be computed using the
Riemann–Roch theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...
). Then, the stack
: