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 curves. It is thus a special case of a
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...
. 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
Fine may refer to:
Characters
* Sylvia Fine (''The Nanny''), Fran's mother on ''The Nanny''
* Officer Fine, a character in ''Tales from the Crypt'', played by Vincent Spano
Legal terms
* Fine (penalty), money to be paid as punishment for an offe ...
and
coarse moduli spaces for the same moduli problem.
The most basic problem is that of moduli of
smooth 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 n ...
. Over the
field of
complex numbers these correspond precisely to
compact Riemann surfaces 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
A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...
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 and
David Mumford,
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 curve
In algebraic geometry, a stable curve is an algebraic curve that is asymptotically stable in the sense of geometric invariant theory.
This is equivalent to the condition that it is a complete connected curve whose only singularities are ordinary ...
s in the
Hilbert scheme
:
of tri-canonically embedded curves (from the embedding of the very ample
for every curve) which have
Hilbert polynomial
In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homoge ...
(note: this can be computed using the
Riemann–Roch theorem). Then, the stack
: