Sharp bounds for higher linear syzygies and classifications of projective varieties (Q2255288)

The paper concerns upper bounds of higher linear syzygies, i.e., graded Betti numbers in the first linear strand of the minimal free resolutions of projective varieties in arbitrary characteristic. Let \(X\) be any nondegenerate variety (i.e. irreducible and reduced closed subscheme) over any field \(k\) of dimension \(n\) and of \(\mathrm{codim}(X, {\mathbb{P}}^N) = e\). Let \(R := k[x_0,\ldots, x_N]\) be the coordinate ring of \({\mathbb{P}}^N\) and \(R_X := R/I_X\) be also the one of \(X\). The graded Betti numbers of \(X\) are defined by \(\beta_{p,q}(X) := \dim_k \mathrm{Tor}^R_p (R_X, k)_{p+q}\). The Betti table of \(X\), \({\mathbf B}(X)\) consists of these graded Betti numbers of \(X\). The subcomplex (or the corresponding part of table) represented by Betti numbers \(\beta_{1,1}\ldots, \beta_{b-1,1}\) in the second row of \({\mathbf B}(X)\) is called the (first) linear strand of \({\mathbf B}(X)\). The (homological) index to which the resolution admits only linear syzygies is denoted by \(a(X)\) and the first index from which there exists no more linear syzygy by \(b(X)\). Then, the linear strand of the minimal free resolution of \(R_X\) can be characterized by these invariants \(a(X)\) and \(b(X)\). \( X\) is a variety of minimal degree (abbr. VMD) if \(d = e + 1\), where \(d = \deg(X)\), degree of \(X\). \(X\) is called of next-to-minimal degree when \(d = e+2\). Furthermore, in this paper, the authors say that \(X\) is a del Pezzo variety if \(X\) is arithmetically Cohen-Macaulay (abbr. ACM) and of next-to-minimal degree. Using Castelnuovo and Fano's results, we have feasible bounds for \(\beta_{1,1}\). But when we move on higher \(p\)'s, it is not so feasible to handle higher linear syzygies directly as to manipulate them in case of \(p\) being very low (e.g. considering generators, their relations, and so on). In this paper, the authors introduce a useful way to treat higher linear syzygies in a quite effective manner, that is Projecting higher linear syzygies of \(X\) to those of its projected image \(X_q\). The main result is Theorem 1.1 whose proof is in section 3 (see Theorem 3.1). Section 2 is devoted to recall Partial Elimination Ideals (PEIs) theory, to give account for its relevance to the theory of projections of projective schemes briefly, and to introduce a new framework in which one can study syzygies of projective subschemes using PEIs theory and reduction method via inner projections. In section 3 the authors give proofs of the main results and add a remark which give some inspiration on how to carry out the computations of Betti numbers using projections. In section 4 they treat next-to-extremal case which is a natural generalization of Fano's classical theorem. In section 5, they give examples and questions to improve the results into more general categories.