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.
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.
