Nonnormal del Pezzo surfaces (Q1890539)

Throughout this paper, a del Pezzo surface is by definition a connected, 2-dimensional, projective \(k\)-scheme \(X,{\mathcal O}_X(1)\) that is Gorenstein and anticanonically polarised; in other words, \(X\) is Cohen-Macaulay, and the dualising sheaf is invertible and antiample: \(\omega_X\cong{\mathcal O}_X(-1)\). The main motivation for the present study was proposition 3.9 of a paper by \textit{S. Mori} [Ann. Math., II. Ser. 116, 133-176 (1982; Zbl 0557.14021)], where the statement that an irreducible del Pezzo surface \(X\) has \(\chi({\mathcal O}_X)\neq 0\) plays an essential role; I reprove here, in particular, Mori's statement that an irreducible de Pezzo surface in characteristic 0 has \(\chi({\mathcal O}_X)=1\). I assume throughout that \(X\) is reduced, but either reducible or (irreducible and) nonnormal; \(\pi:Y\to X\) is the normalisation. \(Y\) has \(r\geq 1\) components, and is marked by an effective Weil divisor \(C\), scheme-theoretically defined by the conductor ideal \(I_{C,Y}\) as a set, \(C\) is the codimension 1 double locus of \(\pi\). By subadjunction (proposition 2.3), it is well known that the canonical Weil divisor of \(Y\) is \(K_Y=\pi^* K_X-C\), so that \[ -K_Y= \pi^*{\mathcal O}_X(1)+C=(\text{ample})+ (\text{effective}). \] It is easy to classify the components \(C\subset Y\) with the property (theorem 1.1) in terms of scrolls and \(\mathbb{P}^2\), and in particular each connected component of \(C\), \({\mathcal O}_C(1)\) must be isomorphic to a plane conic. Write \(D\subset X\) for the subscheme defines by the conductor ideal \(I_{D,X}={\mathcal C}=\text{Ann} (\pi_*{\mathcal O}_Y/{\mathcal O}_X)\subset{\mathcal O}_X\), and \(\varphi:C\to D\) for the restriction of \(\pi\). Thus \(X\) is obtained by glueing together one or more components \(C_i\subset Y_i\) along a morphism \(\varphi:C=\amalg C_i\to D\). The Cohen-Macaulay or \(S_2\) condition for \(X\) is easy: \(\varphi_*{\mathcal O}_C/{\mathcal O}_D\) must have no sections supported at points, so that the glueing is entirely determined in codimension 1. The combinatorics of the glueing also turns out to be straightforward; in particular, if \(Y\) is reducible then all the conics \(C_i\) are isomorphic. What makes a local ring Gorenstein? We have not heard the last of this question. The technical crux of this paper is theorem 2.6, which characterises the predualising sheaf \(\omega_X\) of the nonnormal scheme \(X\). This gives necessary and sufficient conditions on \(C\subset Y\) and \(\varphi:C\to D\) for \(X\) to be Gorenstein, for example: \(\omega_Y(C)\) is an invertible \({\mathcal O}_Y\)-module, \(\omega_X\) is Gorenstein in codimension 1 and \(\omega_C\) has an \({\mathcal O}_C\)-basis \(s\in\ker \{\text{Tr}_{C/D}: \varphi_*\omega_C\to \omega_D\}\). Working with the Gorenstein condition is easy if all the \(C_i\) are reduced, when \(X\) has ordinary double points in codimension 1. But in the nonreduced case, the question is quite subtle, and the answer depends on \(\text{char }k\). In any characteristic, \(\text{Sing } X=\Gamma\) is an irreducible curve of degree 1 for the polarisation. If \(\text{char } k=0\) then \(\Gamma\cong \mathbb{P}^1\) and \(D\), \({\mathcal O}_D(1)\) is isomorphic to a first order neighbourhood of \(\mathbb{P}^1\) in \(\mathbb{P}^r\); this is essentially equivalent to the main result \(\chi({\mathcal O}_X)=1\). However, in characteristic \(p\) the curve \(\Gamma\) can have ``wild'' cusps, and \(\chi({\mathcal O}_X)\) can be arbitrarily negative. Thus for nonnormal varieties in characteristic \(p\), Gorenstein is a weaker condition than in characteristic 0. It follows easily that if \(\text{char } k=0\) then \(H^1(X,{\mathcal O}_X(n))=0\), for all \(n\), in particular \(\chi({\mathcal O}_X)=1\), and a general element \(x_0\in H^0(X,{\mathcal O}_X(1))\) is a non-zero divisor for \({\mathcal O}_X\). However, if \(\text{char }k=p\) then \(H^1(X,{\mathcal O}_X)\) can be arbitrarily large; for an irreducible surface \(X\) this happens only when \(X\) has a curve \(\Gamma\) of cusps, and \(\Gamma\) itself has ``wild'' cusps. Conjecture. Assume that \(\text{char }k=0\), and that \(X\) is 1-connected but not reduced; then \(X,{\mathcal O}_X(1)\) is projectively Cohen-Macaulay (that is, \(H^i(X,{\mathcal O}_X(n))=0\) for \(i=1\) and all \(n\), and for \(i=2\) and \(n\geq 0\)), and a general element \(x_0\in H^0(X,{\mathcal O}_X(1))\) is \({\mathcal O}_X\)-regular, so all the projective embedding properties hold as above.