Moduli space of polarized del Pezzo surfaces and its compactification
A non-singular projective surface X over an algebraically closed field k is called a del Pezzo surface, if \(\omega_ X^{-1}\) is ample where \(\omega_ X\) is the canonical sheaf of X. The self-intersection number \(d=(\omega_ X^{-1}\cdot \omega_ X^{-1})\) is called the degree of the del Pezzo surface X. It is well known that \(d\leq 9\) and if \(d=9\), \(X\cong {\mathbb{P}}^ 2\) and if \(d=8\), \(X\cong {\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\) or \({\mathbb{F}}_ 1\), the one point blowing up of \({\mathbb{P}}^ 2\). If \(d\leq 7\), then X is obtained by blowing up \({\mathbb{P}}^ 2\) at (9-d) points in general position. Put \(f:\quad X\to {\mathbb{P}}^ 2,\) the contraction morphism. In this paper the sheaf \(f^*{\mathcal O}_{{\mathbb{P}}^ 2}(1)\) is called the contraction sheaf of X. In {\S} 1 of this paper the author shows that there exists the coarse moduli space \({\mathcal D}{\mathfrak p}_ d\) of pairs (X,\({\mathcal L})\) consisting of a del Pezzo surface X of degree \(d\leq 7\) and its contraction sheaf \({\mathcal L}\). The author constructs the moduli space as the geometric quotient of a certain open set \(U_ d\) of \(Sym^{9-d}({\mathbb{P}}^ 2)\) by the action of PGL(2). In {\S} 2 the author constructs a natural compactification \({\mathfrak p}_ d\) of \({\mathcal D}{\mathfrak p}_ d\) for 1\(\leq d\leq 4\). Note that for \(d=5,6,7\), \({\mathcal D}{\mathfrak p}_ d\) consists of one point. To each point of the boundary of \({\mathfrak p}_ d\), there corresponds an irreducible reduced surface with at worst singular points of type \(A_ 1\), which is isomorphic to the image of a monoidal transformation of \({\mathbb{P}}^ 2\) with center (9-d) points allowing at most double points, where double point means a subscheme defined by a maximal ideal \({\mathcal I}\) in \({\mathcal O}_{{\mathbb{P}}^ 2}\) such that \(_{k}{\mathcal O}_{{\mathbb{P}}^ 2}/{\mathcal I}=2\) as a k-vector space. The compactification is obtained by blowing up the subspace containing \(U_ d\) with the center outside of \(U_ d\) and taking the universal categorical quotient.
scientific article; zbMATH DE number 3847559