On deformations of \(\mathbb{Q}\)-Fano \(3\)-folds (Q2788620)

A normal projective variety \(X\) defined over an algebraically closed field \(k\) of characteristic zero is called a \(\mathbb{Q}\)-Fano variety if it has terminal singularities and \(-K_X\) is an ample \(\mathbb{Q}\)-Cartier divisor. \(\mathbb{Q}\)-Fano varieties are important in the birational classification of algebraic varieties since they are one of the possible outcomes of the minimal model program.NEWLINENEWLINEA \(\mathbb{Q}\)-smoothing of a complex \(\mathbb{Q}\)-Fano threefold \(X\) is a flat morphism \(f : \mathcal{X}\rightarrow \Delta\), where \(\Delta \) is the unit disk, such that \(f^{-1}(0)=X\) and \(f^{-1}(t)\) has cyclic quotient singularities, for \(t \in \Delta -\{0\}\).NEWLINENEWLINELet \(P\in X\) be a three dimensional terminal singularity of index \(r>0\). Then according to the classification of terminal singularities by M. Reid and S. Mori, \((P\in X )\cong (f=0)/\mathbb{Z}_r\), where \(\mathbb{Z}_r\) acts on \(\mathbb{C}^4\) and \(f=0\) is a \(\mathbb{Z}_r\)-semi-invariant polynomial in four variables for that action. The singularity is called ordinary if \(f\) is invariant for the \(\mathbb{Z}_r\)-action.NEWLINENEWLINEThe main result of this paper is that a complex \(\mathbb{Q}\)-Fano threefold \(X\) with only ordinary terminal singularities has a \(\mathbb{Q}\)-smoothing.NEWLINENEWLINEIn order to prove this result, the author shows first that a \(\mathbb{Q}\)-Fano threefold \(X\) has unobstructed deformations. This is proved by using the fact that \(\mathrm{Def}(X)\cong \mathrm{Def}(U)\), where \(U\) is the smooth locus of \(X\). Then the author shows by explicit calculations that the obstruction in \(\mathrm{Ext}_U^2(\Omega_U,\mathcal{O}_U)\) for extending an infinitesimal deformation of \(U\) is zero. By using this result, the proof of the main theorem reduces to the existence of a good first order deformation of \(X\).NEWLINENEWLINEThe second result of this paper is that if the anticanonical linear system \(|-K_X|\) of \(X\) has a member with only DuVal singularities, then \(X\) has a \(\mathbb{Q}\)-smoothing. Hence in this case the condition that the singularities are ordinary is not needed. However, the property that \(|-K_X|\) contains an element with DuVal singularities is special since there are examples of \(\mathbb{Q}\)-Fano threefolds such that \(|-K_X|\) does not have elements with DuVal singularities or that it is even empty.