\(2 n^2\)-inequality for \(cA_1\) points and applications to birational rigidity (Q6559387)

For the theory of birational rigidity the \(4n^2\) inequality plays a central role. It states that if \(p \in X\) is the germ of a smooth threefold singularity and it is a center of non-canonical singularities of the pair \((X, \frac{1}{n}M)\) where \(M\) is a mobile linear system on \(X\) and \(n\) is a positive rational number then for general members \(D_1\), \(D_2\) in \(M\) it holds that \(\mathrm{mult}_{p}(D_1\cdot D_2) > 4n^2.\)\N\NIn the paper under review it is proved the following important result: \N\NLet \(p \in X\) be the germ of a \(\mathrm{cA}_{1}\) singularity. Let \(M\) be a mobile linear system on \(X\) and let \(n\) be a positive rational number. If \(p\) is a center of non-canonical singularities of the pair \((X, \frac{1}{n}M)\), then for general members \(D_1\), \(D_2\) in \(M\) we have \N\[\N\mathrm{mult}_{p}(D_1\cdot D_2) > 2n^2\N\]\NThe \(2n^2\) inequality has important applications to the birational rigidity of sextic double solids, Fano-weighted complete intersections, and del Pezzo fibrations of degree \(1\) in the context of varieties with \(\mathrm{cA}_1\) points, as shown in the paper. We think it will have more applications as it has the \(4n^2\) inequality.