Abstract. We prove a general embedding theorem for Cohen-Macaulay curves (possibly nonreduced), and deduce a cheap proof of the standard results on pluricanonical embeddings of surfaces, assuming vanishing H 1 (2KX ) = 0. §1. IntroductionLet C be a curve over an algebraically closed field k of characteristic p ≥ 0, and H a Cartier divisor on C. We assume that C is projective and Cohen-Macaulay (but possibly reducible or nonreduced). Write HC = deg O C (H) for the degree of H, p a C = 1 − χ(O C ) for the arithmetic genus of C, and ω C for the dualising sheaf (see [Ha], Chap. III, §7).Our first result is the following. (A cluster Z of degree deg Z = r is simply a 0-dimensional subscheme with length O Z = dim k O Z = r; a curve B is generically Gorenstein if, outside a finite set, ω B is locally isomorphic to O B . The remaining definitions and notation are explained below.) Theorem 1.1. (Curve embedding theorem) H is very ample on C if for every generically Gorenstein subcurve B ⊂ C, either 1. HB ≥ 2p a B + 1, or 2. HB ≥ 2p a B, and there does not exist a cluster Z ⊂ B of degree 2 such thatMore generally, suppose that Z ⊂ C is a cluster (of any degree) such that the restriction for every generically Gorenstein subcurve B ⊂ C. Theorem 1.1 is well known for nonsingular curves C. Although particular cases were proved in, it was clear that the result was more general. In discussion after a lecture on the Gorenstein case by the first author at the May 1994 Lisboa AGE meeting, the fourth author pointed out the above result, where C is only assumed to be a pure 1-dimensional scheme. For divisors on a nonsingular surface, Mendes Lopes [ML] has obtained results analogous to Theorem 1.1 and to Theorem 3.6. We apply these ideas to the canonical map of a Gorenstein curve in §3.The proof of Theorem 1.1 is based on two ideas from Serre and Grothendieck duality:(a) we use Serre duality in its "raw" formwhere d denotes duality of vector spaces.(b) If O C has nilpotents, a nonzero map ϕ: F → ω C is not necessarily generically onto; however (because we are Hom'ming into ω C ), duality gives an automatic factorisation of ϕ of the formvia a purely 1-dimensional subscheme B ⊂ C, where F |B → ω B is generically onto. See Lemma 2.4 for details.Since our main result might otherwise seem somewhat abstract and useless, we motivate it by giving a short proof in §4, following the methods of [C-F], of the following result essentially due to Bombieri (when char k = 0) and to Ekedahl and Shepherd-Barron in general. Recall that a canonical surface (or canonical model of a surface of general type) is a surface with at worst Du Val singularities and K X ample. The remaining notation and definitions are explained below. Here H 1 (2K X ) = 0 follows at once in characteristic 0 from Kodaira vanishing or Mumford's vanishing theorem. One can also get around the assumption H 1 (2K X ) = 0 in characteristic p > 0 (see [Ek] or [S-B]). In fact Ekedahl's analysis (see [Ek, Theorem II.1.7]) shows that H 1 (2K S ) = 0 is only possible in a very special case,...
We classify log-canonical pairs (X, ∆) of dimension two with KX +∆ an ample Cartier divisor with (KX + ∆) 2 = 1, giving some applications to stable surfaces with K 2 = 1. A rough classification is also given in the case ∆ = 0.
In this paper we consider Gorenstein stable surfaces with KX2=1 and positive geometric genus. Extending classical results, we show that such surfaces admit a simple description as weighted complete intersection. We exhibit a wealth of surfaces of all possible Kodaira dimensions that occur as normalisations of Gorenstein stable surfaces with KX2=1; for pg=2 this leads to a rough stratification of the moduli space. Explicit non‐Gorenstein examples show that we need further techniques to understand all possible degenerations.
We classify Gorenstein stable numerical Godeaux surfaces with worse than canonical singularities and compute their fundamental groups.
We classify normal stable surfaces with $$K_X^2 = 1$$ K X 2 = 1 , $$p_g = 2$$ p g = 2 and $$q=0$$ q = 0 with a unique singular point which is a non-canonical T-singularity, thus exhibiting two divisors in the main component and a new irreducible component of the moduli space of stable surfaces $${{\overline{{{\mathfrak {M}}}}}}_{1,3}$$ M ¯ 1 , 3 .
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