Fano 3-folds with divisible anticanonical class
Let \(X\) be a Fano \(3\)-fold, i.e. a normal projective \(3\)-fold with \(\mathbb Q\)-factorial singularities such that \(-K_X\) is ample and \(\rho(X)=1\). The index \(f\) of \(X\) is the biggest integer \(f>0\) such that \(-K_X=fA\) for some Weil divisor \(A\). The divisor \(A\) is a primitive ample divisor and there is a corresponding graded ring \(R(X,A)=\bigoplus _{n\geq 0} H^0(X,\mathcal O _X(nA))\). In the paper under review, it is shown that there are at most 1,492 power series corresponding to the Hilbert series of \(R(X,A)\) where \(X\) is a Fano \(3\)-fold of index \(2\) with \(-K_X=2A\). From this classification it then follows that \((-K_X)^3\geq 8/165\) and that if \(X\) is a Fano 3-fold of index at least \(2\), then \(H^0(X,-K_X)\neq 0\). The case in which \(f\geq 3\) had been previously established in ``Computing Fano 3-folds of index \(\geq 3\)'' [to appear in Jpn. J. Indust. Appl. Math. 11 (2007)].
scientific article; zbMATH DE number 5162386
scientific article