Assumption of genus at least $2$ for stable curves In the article "The irreducibility of the space of curves of given genus" by Deligne, Mumford, the definition of stable curves start with the assumption that the genus is at least $2$. Why is this necessary?
Any reference or idea will be very helpful.
 A: In this paper of Deligne and Mumford they define the notion of a stack. This is a generalization of the notion of a scheme, introduced to accomodate moduli problems where the objects parametrized have automorphisms. For a (non-)example, the Grassmannian is the moduli space parametrizing $k$-dimensional subspaces of a fixed vector space $V$; the fact that $V$ is fixed menas that such a subspace has no automorphisms, which explains morally why the moduli space is a scheme in this case. However, curves often do have nontrivial automorphisms, and for this reason the moduli functor of curves is a stack, not a scheme. 
The notion of stack introduced in their paper is what's now called a Deligne-Mumford stack. These are the stacks closest to ordinary schemes. The important property of a Deligne-Mumford stack is that the automorphism group of any object is finite. Now the automorphism group of a smooth genus $g$ curve is finite if and only if $g \geq 2$, and this is why they make this assumption. 
It is common to also consider curves with marked points. These are parametrized by a moduli space $M_{g,n}$. More generally, a curve of genus $g$ with $n$ markings has finite automorphism group if and only if $2g-2+n >0$, and the stacks $M_{g,n}$ are of Deligne-Mumford type if and only if this inequality is satisfied.
There is a more general notion of Artin stack, which accomodates also infinite stabilizer groups. If you don't mind working with Artin stacks then the moduli spaces $M_{g,n}$ for $2g-2+n \leq 0$ are perfectly sensible objects to consider, too.
