MATHEMATICS SINGULARITIES OF THE MODULI SCHEME FOR CURVES OF GENUS THREE BY FRANS OORT (Communicated by Prof. J. P. MURRE at the meeting of September 28, 1974) Let M, be the (coarse) moduli scheme over Z for smooth, irreducible algebraic curves of genus g (cf. [6], Coroll. 7.14). THEOREM. Let lc be a field, P a geometric point of MS 8 k, and CP a curve over some algebraically closed field corresponding with P. Suppose Cp is hyperelliptic. The point P is singular on MS @ I% if and only if /Aut (C)l > 2. REMARK. This was proved in case char (Ic)=O by RAUCH (cf. [lo], Th. 1. ii), and conjectured by POPP (cf. [9], p. 91) in the general case. This result is the last detail in the work by Rauch, Popp and others in describing the singularities of coarse moduli schemes for curves: Ml s Al is non-singular ; for Mz, cf. [4], Theorem 4 (note that “the variety of moduli” constructed by Igusa for g = 2 is isomorphic with the scheme Ma constructed by MUMBORD, cf. [8], Fact 10) ; for g 2 4, or g= 3 and C not hyperelliptic, P is singular on M, if and only if Cp has non-trivial automorphisms (cf. [lo], and [9]). LEMMA 1. Let Mg be as above, let A, =A,,l,i be the coarse moduli scheme over Z for principally polarized abelian varieties with level one structure (cf. [6], Coroll. 7.14 and Th. 7.10). If g= 1, 2, 3, the morphism (cf. [S], 7.4) j: M,+A, is an open immersion. LEMMA 2. Let k: be a field, V an irreducible subscheme of MZ @I k, and P the generic point of Tr; let C= Cp be a curve (over some algebraically closed field) corresponding with P. Assume C admits a non-trivial auto- morphism t; in case C is hyperelliptic, assume moreover order (z) # 2 or C/z c/g Pi (i.e. z is not the involution which makes 0 hyperelliptic). Then dim Vg4.