Sharp bounds for higher linear syzygies and classifications of projective varieties

Abstract

In the present paper, we consider 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. For this purpose, we first remind 'Partial Elimination Ideals (PEIs)' theory and introduce a new framework in which one can study the syzygies of embedded projective schemes well using PEIs theory and the reduction method via inner projections.Next we establish fundamental inequalities which govern the relations between the graded Betti numbers in the first linear strand of an algebraic set X and those of its inner projection Xq. Using these results, we obtain some natural sharp upper bounds for higher linear syzygies of any nondegenerate projective variety in terms of the codimension with respect to its own embedding and classify what the extremal case and next-to-extremal case are. This is a generalization of Castelnuovo and Fano's results on the number of quadrics containing a given variety and another characterization of varieties of minimal degree and del Pezzo varieties from the viewpoint of 'syzygies'. Note that our method could be also applied to get similar results for more general categories (e.g. connected in codimension one algebraic sets).

On the first non-trivial strand of syzygies of projective schemes and Condition ${\mathrm ND}(l)$

“…A few years ago, Han and Kwak developed an inner projection method to compare syzygies of X with those of its projections by using mapping cone and partial elimination ideal theory. As applications, over any algebraically closed field k of arbitrary characteristic, they proved the sharp upper bounds on the ranks of higher linear syzygies by quadratic equations, and characterized the extremal and next-to-extremal cases, which generalized the results of Castelnuovo and Fano [HK15]:…”

“…Since the foundational paper on syzygy computation by Green ([Gre84]), there has been a great deal of interest and progress in understanding the structure of the Betti tables of algebraic varieties during the past decades. In particular, the first non-trivial linear strand starting from quadratic equations has been intensively studied by several authors ( [Cas1893], [Gre84], [GL88], [EGHP05,EGHP06], [EL15], [HK15] etc. ).…”

A matryoshka structure of higher secant varieties and the generalized Bronowski's conjecture

“…In fact, it is proved in Corollary 2.14 that β p,q (S q (X )) ≤ p+q−1 q e+q p+q for every p. Thus, among all q-secant varieties, q-secant varieties of minimal degree are characterized to have the maximal Betti number β p,q (S q (X )) = p+q−1 q e+q p+q for some 1 ≤ p ≤ e. For q = 1, it is classical and was proved by Castelnuovo, Eisenbud-Green-Hulek-Popescu and Han-Kwak ( [8], [15], [27]). Furthermore, in terms of determinantal presentation, we describe higher secant varieties of minimal degree.…”

“…Figure 2: The Betti table of del Pezzo q-secant varieties For q = 1, it is classical and was proved by Fano, and Han-Kwak ( [17], [27]). It was also proved for higher secant varieties to elliptic normal curves due to Bothmer and Hulek, or Fisher (see [3], [18]).…”

Linear normality of general linear sections and some graded Betti numbers of 3-regular projective schemes

“…see [4, chap. 8,9] for an overview of its connection toward Green and Green-Lazarsfeld conjectures of curves or see [12] for its relevance to classification of varieties of small degree). For 3 rd -quadrant, there is a nice geometric interpretation related to the existence of a degenerate secant plane X ∩ Λ, which is a finite scheme with length(X ∩ Λ) > dim Λ+ 1 for a linear subspace Λ of dimension ≤ e (see e.g.…”