In algebraic geometry, a moduli space of curves is a space whose points correspond to isomorphism classes of algebraic curves. The term "modulus" was introduced for this purpose by Bernhard Riemann, and means "parameter"; thus a "moduli space" means a space giving parameters that specify all of the curves of a given kind. With a moduli space, instead of studying one curve at a time, one studies all curves of a given kind as members of a single geometric family. The moduli space of curves (of a given kind) is a special case of the more general notion moduli space, which gives a parameter space for other kinds of objects (curves, surfaces, etc). Different choices of conditions lead to different moduli spaces. For example, one may fix the genus, allow only smooth or also certain singular curves, or include marked points. Depending on the problem, the moduli object may be constructed as a scheme, an algebraic space, or more naturally as an algebraic stack. In many cases there is both a coarse moduli space, which records isomorphism classes of curves, and a finer stack that also keeps track of their automorphisms. A well-studied example is the moduli of smooth projective curves of genus g {\displaystyle g} . Over the field of complex numbers, these correspond to compact Riemann surfaces. Classically, the (coarse) moduli space of genus g = 1 {\displaystyle g=1} curves having a marked point (elliptic curve groups) is the (classical) modular curve. For g > 1 {\displaystyle g>1} , the moduli stack of smooth curves is denoted M g {\displaystyle {\mathcal {M}}_{g}} , and its compactification by stable nodal curves is denoted M ¯ g {\displaystyle {\overline {\mathcal {M}}}_{g}} . These spaces and stacks play a central role in algebraic geometry, Teichmüller theory, and the theory of modular forms.
Moduli stacks of stable curves The moduli stack M g {\displaystyle {\mathcal {M}}_{g}} classifies families of smooth projective curves, together with their isomorphisms. When g > 1 {\displaystyle g>1} , 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 M ¯ g {\displaystyle {\overline {\mathcal {M}}}_{g}} . Both moduli stacks carry universal families of curves. Both stacks above have dimension 3 g − 3 {\displaystyle 3g-3} ; hence a stable nodal curve can be completely specified by choosing the values of 3 g − 3 {\displaystyle 3g-3} parameters, when g > 1 {\displaystyle g>1} . In lower genus, one must account for the presence of smooth families of automorphisms, by subtracting their number. There is exactly one equivalence class of complex curves of genus zero, namely the Riemann sphere, and its group of automorphisms is PGL(2). Hence the dimension of M 0 {\displaystyle {\mathcal {M}}_{0}} is equal to
dim ( space of genus 0 curves ) − dim ( group of automorphisms ) = 0 − dim ( P G L ( 2 ) ) = − 3. {\displaystyle {\begin{aligned}\dim({\text{space of genus 0 curves}})-\dim({\text{group of automorphisms}})&=0-\dim(\mathrm {PGL} (2))\\&=-3.\end{aligned}}}
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 M 1 {\displaystyle {\mathcal {M}}_{1}} has dimension 0.
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