2014/02/13 by Ahn, Jeaman, Kwak, Sijong
#Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Primary:14N05 #Secondary:13D02
paper · doi:10.48550/arxiv.1402.3100
For an algebraic set X (union of varieties) embedded in projective space, we say that X satisfies property Nd,p, (d≥ 2) if the i-th syzygies of the homogeneous coordinate ring are generated by elements of degree < d+i for 0≤ i≤ p (see \citeEGHP2 for details). Much attention has been paid to linear syzygies of quadratic schemes (d=2) and their geometric interpretations (cf. \citeAK,\citeEGHP1,\citeHK,\citeGL2,\citeKP). However, not very much is actually known about the case satisfying property N3,p. In this paper, we give a sharp upper bound on the maximal length of a zero-dimensional linear section of X in terms of graded Betti numbers (Theorem 1.2 (a)) when X satisfies property N3,p. In particular, if p is the codimension e of X then the degree of X is less than or equal to \binome+22, and equality holds if and only if X is arithmetically Cohen-Maucalay with 3-linear resolution (Theorem 1.2 (b)). This is a generalization of the results of Eisenbud et al. (\citeEGHP1,EGHP2) to the case of N3,p, (p≤ e).