Divisorial extractions from singular curves in smooth 3-folds (Q2790324)

A projective morphism \(f : Y \rightarrow X\) between two normal varieties with \(\mathbb{Q}\)-factorial singularities is called a divisorial contraction if and only if there exists an irreducible divisor \(E \subset Y\) such that \(\Gamma=f(E)\) has codimension greater or equal than 2 in \(X\), \(Y-E\cong X-\Gamma\), \(-K_Y\) is \(f\)-ample and the relative Picard number \(\rho(Y/X)\) of \(Y\) over \(X\) is one. A divisorial contraction is called terminal if and only if \(X\) and \(Y\) have terminal singularities. According to the minimal model program, terminal three dimensional divisorial contractions form one of the three fundamental classes of birational maps that appear in the birational classification of threefolds. The other two are flips and flops. The explicit classification of these maps is intimately related to the explicit birational classification of threefolds and the birational maps between them.NEWLINENEWLINELet \(\Gamma \subset X\) be a singular curve with non complete intersection singularities in a smooth threefold \(X\). In this paper the author investigates when a three dimensional terminal divisorial contraction \(f : Y \rightarrow X\) exists such that \(f\) contracts an irreducible divisor \(E\) of \(Y\) onto \(\Gamma\). This problem is local around the singular points of \(\Gamma\) and therefore it is reduced to the case when \(P\in\Gamma \subset X\) is the germ of a smooth threefold at a singular non complete intersection point \(P\in \Gamma\). Let \(S \subset X\) be the general hyperplane section of \(X\) containing \(\Gamma\). According to Reid's general elephant conjecture, if a terminal contraction exists then \(S\) has DuVal singularities (this is known to be true if a terminal contraction exists such that \(f^{-1}(P)\) is irreducible [\textit{J. Kollár} and \textit{S. Mori}, J. Am. Math. Soc. 5, No. 3, 533--703 (1992; Zbl 0773.14004)]). Based on this observation the problem is reduced to the case when \(S\) has DuVal singularities.NEWLINENEWLINEThe main results of the paper are the following: {\parindent=6mm \begin{itemize}\item[1.] If \(P\in S\) is a DuVal singularity of type \(D_{2n}\) or \(E_7\) then there does not exist a divisorial contraction \(f : Y \rightarrow X\) contracting an irreducible divisor onto \(\Gamma\). \item[2.] If \(S\) is of type \(E_6\) then the author gives a criterion for the existence of a contraction with respect to the position of \(\Gamma\) in the dual graph of the minimal resolution of \(S\). \item[3.] If \(S\) is of type \(A_1\) or \(A_2\) then the author obtains criteria for the existence of a contraction with respect to the form of the local equations of \(\Gamma\) in \(X\). NEWLINENEWLINE\end{itemize}}NEWLINENEWLINEIn order to prove these results the author follows the following method. The first step is to obtain normal forms for the equations of \(\Gamma \subset S \subset X\). Then let \(g : Y^{\prime} \rightarrow X\) be the blow up of \(X\) along \(\Gamma\). There are two irreducible \(g\)-exceptional divisors. A divisor \(E^{\prime}\) over \(\Gamma\) and a divisor \(F=\mathbb{P}^2\) over the singularity \(P\) of \(\Gamma\). By using the Kustin-Miller unprojection method developed by \textit{S. A. Papadakis} and \textit{M. Reid} [J. Algebr. Geom. 13, No. 3, 563--577 (2004; Zbl 1071.14047)], there exists a birational map \(\phi : Y^{\prime}\dasharrow Y\) over \(X\) contracting \(F\), and thus producing a birational map \(f : Y \rightarrow X\) contracting the birational transform \(E\) of \(E^{\prime}\) onto \(\Gamma\). Then by explicitly writing down the equations of \(Y\), the author studies the singularities of \(Y\). A terminal contraction exists if and only if \(Y\) has terminal singularities.