Moduli Stacks of Stable Curves
The moduli stack classifies families of smooth projective curves, together with their isomorphisms. When 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 . 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 3g-3 parameters, when g > 1. 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
- dim(space of genus zero curves) - dim(group of automorphisms) = 0 - dim(PGL(2)) = -3.
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.
Read more about this topic: Moduli Of Algebraic Curves
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