Rational surfaces with a large group of automorphisms (Q2892820)

Let \(X\) be a smooth projective surface over an algebraically closed field \(K\) and consider the natural action of \(\text{Aut}(X)\) on \(\text{Pic}(X)\); denote by \(\text{Aut}(X)^*\) the image of \(\text{Aut}(X)\) in the orthogonal group \(\text{O}(\text{Pic}(X))\).NEWLINENEWLINEWhen \(X\) is rational and \(\text{Aut}(X)^*\) is infinite, it is known that \(X\) is the blow-up of \({\mathbb P}^2\) along a point set \({\mathcal P}_n=\{p_1,\ldots,p_n\}\), with \(n\geq 9\). As explained in the paper under review, in this situation one has a Weyl group \(W_X\) associated to a root system for \(\text{O}(K_X^\perp)\) and it may be shown \(\text{Aut}(X)^*\subset \text{O}(K_X^\perp)\). A point set \({\mathcal P}_n\) is said to be \textit{Cremona special} if (\(n\geq 9\) and) \(\text{Aut}(X)^*\) has finite index in \(W_X\). By blowing up Cremona point sets, for a fixed \(n\), one obtains surfaces with largest possible discrete automorphism group among the rational surfaces with Picard group of rank \(n+1\).NEWLINENEWLINEThe main result of the paper under review is the classification of Cremona special sets. More precisely, the authors of that paper prove that if \({\mathcal P}_n\) is Cremona special, then the blow-up of \({\mathbb P}^2\) along \({\mathcal P}_n\) does not contain \((-2)\)-curves and one of the following situations occurs:NEWLINENEWLINE\noindent (a)\ \(n=9\) and \({\mathcal P}_9\) is the base-point set of a pencil of degree \(3m\) curves for some \(m\geq 1\) (Halphen sets)NEWLINENEWLINE\noindent (b)\ \(n=10\), nine of the ten points in \({\mathcal P}_{10}\) are as in (a), for \(m=2\), and the other one is a singular point of an irreducible non multiple element of the corresponding pencil (Coble sets).NEWLINENEWLINE\noindent (c)\ \(n\geq 10\), \(\text{Char}(K)>0\), and \({\mathcal P}_n\) consists of general nonsingular points of an irreducible cuspidal cubic curve (Harbourne sets). \smallskip This answers to a question raised by Arthur Coble in 1928.