On the locus of curves with automorphisms (Q1105652)

Let \(S_ g\) be the locus of curves with a nontrivial automorphism in the moduli space \(M_ g\) of curves of genus \(g\geq 3\) (for \(g\geq 4\), \(S_ g\) is also the singular locus of \(M_ g)\). The author describes the irreducible components of \(S_ g\) in the following way: if C is a smooth curve of genus \(g\) with an automorphism \(\tau\) of prime order p, the quotient curve \(X=C/<\tau >\) is smooth of genus g', and \(C\to X\) is ramified at \(n\geq 0\) points \(q_ 1,...,q_ n\) in C, such that the Riemann-Hurwitz relation \(2g-2=p\cdot (2g'-2)+n(p-1)\) is satisfied. If \(z_ i\) is a local parameter at \(q_ i\) and \(\zeta\) a primitive p-th root of unity, then there are unique integers \(a_ i\leq p-1\) such that \(z_ i(P)=\zeta^{a_ i}z_ i(\tau (P))\). The space \(S_ g(p,g';a_ 1,...,a_ n)\) of all such coverings is an irreducible subspace of \(M_ g\) of dimension \(3g'-3+n\), and with a few exceptions which are explicitly described in the text, all of them are irreducible components of \(S_ g\), and conversely each component of \(S_ g\) is of the form \(S_ g(p,g';a_ 1,...,a_ n).\) The main technical lemma asserts that in the situation above, Aut(C/X) is a normal subgroup of Aut(C) (with exceptions \(g'=0\), \(n=3\) or \(g'=1\), \(n=2)\); its proof needs some deformation theory. The paper concludes with a table of the irreducible components of \(S_ g\) for \(g\leq 50\).