Corti, Alessio Del Pezzo surfaces over Dedekind schemes. (English) Zbl 0902.14026 Ann. Math. (2) 144, No. 3, 641-683 (1996). Definition. Let \(K\) be a field. A Del Pezzo surface is a Gorenstein surface \(X_K\) over \(K\), with ample anticanonical bundle.The most important discrete invariant of a Del Pezzo surface is its degree, the self-intersection number of the canonical class. It is well known that \(1\leq\deg\leq 9\). – We fix a Dedekind scheme \(S\) with fraction field \(K \). We denote by \(\eta= \text{Spec} K\) the generic point of \(S\). We study the following problem(*) Given a smooth Del Pezzo surface \(X_K\) over \(K\), find a “nice” integral model of \(X_K\), i.e., a nice scheme \(X\) over \(S\) such that \(X_\eta =X_K\).There are two sources of motivation to study (*), geometry and arithmetic. In the arithmetic context, \(K\) is a number field, my own motivation to consider (*) comes from the birational classification theory of 3-folds, namely the study of Mori fiber spaces of dimension 3.Definition. A Mori fiber space is a projective morphism \(f:X\to T\), where \(X\) is a projective variety with \(\mathbb{Q}\)-factorial terminal singularities and \(f:X\to T\) is an extremal contraction of fibering type. This means that: 1. \(f\) has connected fibers, \(T\) is normal, and \(\dim(T) <\dim (X)\) (in general, \(f\) is not equidimensional).2. \(-K_X\) is \(f\)-ample and the rank of the relative Picard group is 1: \(\rho (X/T) =\rho (X)- \rho(T) =1\).The requirements that \(X\) be \(\mathbb{Q}\)-factorial and \(\rho (X/T) =1\) are crucial. – When \(X\) is 3-dimensional, Mori fiber spaces come in three kinds, according to the dimension of the base space \(T\):\(\dim (T)=0\). In this case \(X\) is a \(\mathbb{Q}\)-Fano 3-fold with Picard number \(\rho(X) =1\).\(\dim (T)=1\). \(X\to T\) is a (flat) Del Pezzo fibration, in the sense that the generic fiber is a smooth Del Pezzo surface. This is the case of most interest to us.\(\dim(T) =2\). \(X\to T\) is a conic bundle, in the sense that the generic fiber is a smooth conic. \(f\) is not necessarily flat, but all fibers are curves.The minimal model theorem for 3-folds, which, conjecturally, holds over an arbitrary base scheme, provides a first answer to (*) at least if \(\rho (X_K)=1\), namely a Mori fiber space \(X\to T\) of \(\dim (T)=1\), that is, one which is a Del Pezzo fibration. This solution is not satisfactory: The point is that there are 3-fold Mori fiber spaces over a curve with singular points of arbitrarily large index. In particular, the minimal embedding dimension of these models can be arbitrarily large, which is quite unpleasant, especially for arithmetic purposes. This phenomenon is very common, even when the generic fiber is \(\mathbb{P}^2\).Gluing together models over the spectra of the local rings of \(S\), it is enough to answer (*) when \(S=\text{Spec} {\mathfrak O}\) is the spectrum of a discrete valuation ring \({\mathfrak O}\).Let \(X_K\) be a smooth Del Pezzo surface of degree \(d\) over \(K\). A model of \(X_K\) is a scheme \(X\), defined and flat over \({\mathfrak O}\), such that \(X_\eta= X_K\).Assume \(d\geq 3\). A standard model of \(X_K\) is a model \(X\) of \(X_K\) satisfying the following conditions:\(X\) has terminal singularities of index 1;the central fiber \(X_0\) is reduced and irreducible, in particular \(X_0\) is a Gorenstein Del Pezzo surface;the anticanonical system \(-K_X\) is very ample and defines an embedding \(X \subset \mathbb{P}^d_{\mathfrak O}\).Theorem. If \(d\geq 3\), then a standard model for \(X_K\) exists.If \(d=1\) or 2, then models with terminal singularities of index 1 do not always exist.Definition. Assume \(d\leq 2\). A model \(X\) of \(X_K\) is a standard model if:\(X\) has terminal singularities;the central fiber \(X_0\) is reduced and irreducible;if \(d=2\), \(-2K_X\) is Cartier and very ample;if \(d=1\), \(-aK_X\) is Cartier and very ample for \(a\in\{1, 2, 3, 4, 6\}\). Theorem. If \(d=2\), then a standard model of \(X_K\) exists.Conjecture. If \(d=1\), then a standard model of \(X_K\) exists. Cited in 2 ReviewsCited in 20 Documents MSC: 14J25 Special surfaces 14G27 Other nonalgebraically closed ground fields in algebraic geometry 14J10 Families, moduli, classification: algebraic theory Keywords:Fano 3-fold; Del Pezzo surface; ample anticanonical bundle; Dedekind scheme; Mori fiber space; terminal singularities; extremal contraction; Del Pezzo fibration; smooth Del Pezzo surface; standard model × Cite Format Result Cite Review PDF Full Text: DOI arXiv