Redundant blow-ups of rational surfaces with big anticanonical divisor
The authors prove the existence and provide a complete classification of the so-called redundant blow-ups of rational surfaces having big anticanonical divisor. This classification is motivated by a paper of \textit{F. Sakai} [Math. Ann. 269, 389--410 (1984; Zbl 0533.14016)], where the author shows that the anticanonical morphism \(f: S \rightarrow \overline{S}\) factors through the minimal resolution of a del Pezzo surface with only rational singularities followed by a sequence of redundant blow-ups. For a normal projective surface \(S\) with \(-K_{S}\) pseudoeffective we denote by \(-K_{S} = P + N\) its Zariski decomposition. We say that \(p \in S\) is a redundant point if \(\mathrm{mult}_{p}(N) \geq 1\) and we called the blow up \(f: \tilde{S} \rightarrow S\) at a redundant point \(p\) as a redundant blow-up. In order to classify all such redundant blow-ups one needs only to read off the information about redundant points on the surface \(S\). The first result of this note is the following. Theorem 1. Let \(\overline{S}\) be a normal projective surface with nef anticanonical \(\mathbb{Q}\)-Cartier divisors and let \(\tilde{S}\) be a surface obtained by a sequence of redundant blow-ups from the minimal resolution \(S\) of \(\overline{S}\). We have the following. {\parindent=6mm \begin{itemize} \item[1)] The number of surfaces obtained by a sequence of redundant blow-ups from \(S\) is finite if and only if \(\overline{S}\) contains at worst log terminal singularities. \item [2)] If \(\overline{S}\) contains at worst log terminal singularities, then every redundant point on \(\tilde{S}\) lies on the intersection points of the two curves contracted by the morphism \(h: \tilde{S} \rightarrow S\). 3) If \(\overline{S}\) contains a non-log terminal singularity, then there is a curve in \(\tilde{S}\) contracted by the morphism \(h: \tilde{S} \rightarrow \overline{S}\) such that every point lying on this curve is a redundant point. \end{itemize}} The second result of this article is devoted to the existence of redundant points, which can be determined from the singularity types of \(\overline{S}\). {Theorem 2.} Let \(\overline{S}\) be a normal projective surface with nef anticanonical \(\mathbb{Q}\)-Cartier divisor and let \(g: S \rightarrow \overline{S}\) be its minimal resolution. Then \(S\) has no redundant points if and only if \(\overline{S}\) contains at worst canonical singularities or log terminal singularities whose dual graphs are described by the following sequences (which means that all dual graphs are linear) {\parindent=6mm \begin{itemize} \item[1)] \((-2,-2,\dots, -2, -3)\), where the number of \(-2\)'s is equal to \(\alpha \geq 1\), \item [2)] \((-2,-2,-3,-2)\), \item [3)] \((-2,-3,-2)\), \item [4)] \((-2,-4)\), \item [5)] \((-n)\) with \(n \geq 3\). \end{itemize}} The last result of this article gives a negative answer to the question posed by Testa, Várilly-Alvarado and Velasco [\textit{D. Testa} et al., Math. Ann. 351, No. 1, 95--107 (2011; Zbl 1226.14049)], i.e. whether every big anticanonical rational surface is the minimal resolution of a del Pezzo surface with only rational singularities. Theorem 3. For each \(n \geq 10\) there exists a big anticanonical rational surface of Picard number \(n\) which is not a minimal resolution of a del Pezzo surface with only rational singularities. Moreover, in the last part the authors present several very instructive examples which are also important in the context of the theory of Zariski decompositions and negative curves on algebraic surfaces. The article is almost self-contained and pleasant to read.
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