Some results on rational surfaces and Fano varieties (Q2744701)

The main object of the paper under review is a very ample line bundle \(L\) on a projective variety \(X\) (the focus is on two cases where \(X\) is a rational surface or a Fano variety). The authors are interested in the following property of \(L\): Given a positive integer \(p\), \(L\) is said to satisfy property \(N_p\) if \(|L|\) embeds \(X\) as a projectively normal variety, the homogeneous ideal \(I\) of the image of \(X\) embedded by \(|L|\) is generated by quadratic equations, and the matrices in the minimal graded free resolution of the homogeneous coordinate ring \(S/I\) of an embedded variety have linear entries from the second to the \(p\)-th step. NEWLINENEWLINENEWLINEFirst, the authors consider the case where \(X\) is a rational surface. They give a numerical criterion for property \(N_p\) in the case where \(L\) is base-point-free: To satisfy property \(N_p\) it is sufficient (and if \(X\) is anticanonical also necessary) that \(-K_X\cdot L\geq p+3\). This result generalizes a theorem of \textit{B.~Harbourne} [J. Algebra 190, 145-162 (1997; Zbl 0883.14017)]. The next numerical result states that if \(X\) is of degree \(d=K^2_X\geq 1\) and \(L\) has high self-intersection number, then \(K_X+L\) has property \(N_p\) for \(p\) big enough. Finally, the authors relate property \(N_p\) with the ``termination'' of ampleness of \(mK_X+L\). They give sharp bounds (depending on \(K_X\) and \(p\)) for \(m\) such that \(X\) is an anticanonical surface, \(L\) satisfies property \(N_p\) but not property \(N_{p+1}\), and \(mK_X+L\) is not ample. They also study syzygies associated to adjunction bundles: Given ample line bundles \(A_1,\dots ,A_n\) on an anticanonical surface \(X\) of fixed degree, they prove a sharp bound on \(n\) such that \(K_X+A_1+\dots +A_n\) satisfies property \(N_p\). NEWLINENEWLINENEWLINESecond, \(n\)-dimensional Fano varieties of index \(\geq n-1\) and \(n-3\) are considered (the case of index \(n-2\) was treated by the authors in an earlier paper [\textit{F. J. Gallego} and \textit{B. P. Purnaprajna}, J. Pure Appl. Algebra 146, 251-265 (2000; Zbl 1028.14004)]). If \(-K_X=(n-1)H\) for an ample and base-point-free line bundle \(H\) (e.g. if \(X\) is a Fano \(n\)-fold of index \(n-1\)), then \(H\) satisfies property \(N_p\) if and only if \(H^n\geq p+3\). The authors also present some results on Koszul cohomology and syzygies for multiples of \(H\). For a Fano \(n\)-fold of index \(n-3\), they give a criterion for \(nH\) to be very ample and satisfy property \(N_0\) (i.e. to embed \(X\) as a projectively normal variety) and prove a result on higher syzygies of multiples of \(H\).