In this paper, we study the minimal free resolution of a nondegenerate projective variety X⊂Pr when X is contained in a variety Y of minimal degree as a divisor. Such a variety is of interest because of its extremal behavior with respect to various properties. The graded Betti diagram of X has been completely known only when X is arithmetically Cohen‐Macaulay. Our main result in the present paper provides a detailed description of the graded Betti diagram of X for the case where X is not arithmetically Cohen‐Macaulay.

We study projective varieties X ⊂ P r of dimension n ≥ 2, of codimension c ≥ 3 and of degree d ≥ c + 3 that are of maximal sectional regularity, i.e. varieties for which the Castelnuovo-Mumford regularity reg(C) of a general linear curve section is equal to d − c + 1, the maximal possible value (see [10]). As one of the main results we classify all varieties of maximal sectional regularity. If X is a variety of maximal sectional regularity, then either (a) it is a divisor on a rational normal (n + 1)-fold scroll Y ⊂ P n+3 or else (b) there is an n-dimensional linear subspace F ⊂ P r such that X ∩ F ⊂ F is a hypersurface of degree d − c + 1. Moreover, suppose that n = 2 or the characteristic of the ground field is zero. Then in case (b) we obtain a precise description of X as a birational linear projection of a rational normal n-fold scroll.H i (P r , I X (m − i)) = 0 for all i ≥ 1.The m-regularity condition implies the (m + 1)-regularity condition, so that one defines the Castelnuovo-Mumford regularity reg(X) of X as the least integer m such that X is m-regular. It is well known that if X is m-regular then its homogeneous ideal is generated by forms of degree ≤ m. This algebraic implication of m-regularity has an elementary geometric consequence that any (m + 1)-secant line to X should be contained in X. We say that a linear space L ⊂ P r is k-secant to X if length(X ∩ L) := dim k (O P r /I X + I L ) ≥ k.A well known conjecture due to Eisenbud and Goto (see [6]) says thatObviously this conjecture implies the following conjecture (1.2) X has no proper k-secant line if k > d − c + 1. So far the conjecture (1.1) has been proved only for irreducible but not necessarily smooth curves by Gruson-Lazarsfeld-Peskine[10] and for smooth complex surfaces by H. Pinkham[20] and R. Lazarsfeld[14]. Moreover, in [10] the curves in P r whose regularity takes the maximally possible value d − r + 2 are completely classified: they are either

a b s t r a c tIn this article we study the classification of non-normal cubic hypersurfaces over an algebraically closed field K of arbitrary characteristic. Let X ⊂ P r K be an irreducible nonnormal cubic hypersurface. If r ≥ 5, then X is necessarily a cone (Remark 2.3). In view of this fact it suffices to classify irreducible non-normal cubic hypersurfaces X ⊂ P r K for r ≤ 4.We prove that there are precisely five non-normal cubic equations (resp. six non-normal cubic equations) when char K ̸ = 2, 3 (resp. when char K is either 2 or 3), up to projective equivalence. Also we describe the normalization of X in detail.

For a very ample line bundle L on a projective scheme X, let1, be the embedding defined by the complete linear series |L |. In this paper we study the problem how the Castelnuovo-Mumford regularity of ϕ L (X) effects on the defining equations of ϕ L (X) and the syzygies among them. We show that if ϕ L (X) ⊂ PH 0 (X, L) is m-regular, then (X, L ) satisfies property N 2 −m+1 for m 2 m − 2, and (X, L ) satisfies property N for all m − 1.

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