Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture

The Eisenbud-Koh-Stillman Conjecture on Linear Syzygies

It is proved, as was conjectured by Eisenbud-Koh-Stillman, that for a finitely generated graded module $M$ over the symmetric algebra $S(V)$, if the Koszul group ${\cal K}_{p,0}(M,V)\ne 0$, then the set of rank 1 relations in $M_0\otimes V$ has dimension $\ge p$. The method is by using ``exterior minors" to study syzygies of an ideal derived from the Koszul class in the exterior algebra. As a consequence, a conjecture of Lazarsfeld and myself that a set $Z$ of $2r+1-p$ points in ${\bf P}^r$ for which property $N_p$ fails has a subset $Z'$ of at least $2{\rm dim}(L) +2-p$ points lying on a line…

Linear determinantal equations for all projective schemes

We prove that every projective embedding of a connected scheme determined by the complete linear series of a sufficiently ample line bundle is defined by the 2-minors of a 1-generic matrix of linear forms. Extending the work of Eisenbud-Koh-Stillman for integral curves, we also provide effective descriptions for such determinantally presented ample line bundles on products of projective spaces, Gorenstein toric varieties, and smooth n-folds.

Arithmetically Cohen-Macaulay Curves cut out by Quadrics

Addressing a question of M. Stillman, it had been shown by Ein, Eisenbud, and the author that in a projective space of dimension at most 5, every arithmetically Cohen-Macaulay curve which is cut out by quadrics scheme- theoretically also has its homogeneous ideal generated by quadrics. In this note it is shown that this is not the case in higher dimensional spaces.

通过线性形式的四向张量构造切簇的二次方程

Quadratic equations of tangent varieties via four-way tensors of linear forms

该研究利用四向线性形式张量构造切簇的二次方程与线性合冲并推广至高阶情形，拓展了经典行列式方程，证明了相关猜想变体，还从表示论角度诠释了Green-Lazarsfeld类。

High rank linear syzygies on low rank quadrics

We study the linear syzygies of a homogeneous ideal I in a polynomial ring S, focusing on the graded betti numbers b_(i,i+1) = dim_k Tor_i(S/I, k)_(i+1). For a variety X and divisor D with S = Sym(H^0(D)*), what conditions on D ensure that b_(i,i+1) is nonzero? Eisenbud has shown that a decomposition D = A + B such that A and B have at least two sections gives rise to determinantal equations (and corresponding syzygies) in I_X; and conjectured that if I_2 is generated by quadrics of rank at most 4, then the last nonvanishing b_(i,i+1) is a consequence of such equations. We describe obstruction…

A generalization of the Clifford index and determinantal equations for curves and their secant varieties

This is the author's 2008 thesis from the University of Chicago. We generalize the notion of the Clifford index to an arbitrary very ample line bundle and show how it determines when a curve and its various secant varieties have determinantal equations. In particular we establish the scheme theoretic version of a conjecture of Eisenbud-Koh-Stillman. If $L_1$ and $L_2$ are line bundles of degree at least $2g+1+k$, then ${\rm Sec}^j(C)$ is determinantally presented for $j \leq k$ and for $j \leq k+1$ if $L_1 \neq L_2$. We also give a geometric characterization of the standard Clifford index in t…