Resultants and Chow forms via exterior syzygies

By David Eisenbud, Frank-Olaf Schreyer. 2001

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Let W be a vector space of dimension n+ 1 over a field K. The Chow divisor of a k-dimensional variety X in P = P(W ) is the hypersurface, in the Grassmannian Gk+1 of planes of codimension k+1 in P, whose points (over the algebraic closure of K) are the planes that meet X . The Chow form of X is the defining equation of the Chow divisor. For example, the resultant of k+ 1 forms of degree e in k + 1 variables is the Chow form of P embedded by the e-th Veronese mapping in P with n = ( k+e k ) − 1. More generally, the Chow divisor of a k-cycle ∑ i ni[Vi] on projective space is defined to be ∑ i niDi, where Di is the Chow divisor of Vi. The Chow divisor of a sheaf F with k-dimensional support is the Chow divisor of the associated k-cycle of F . In this paper we will give a new expression for the Chow divisor and apply it to give explicit formulas in many new cases. Starting with a sheaf F on P, we use exterior algebra methods to define a canonical and effectively computable Chow complex of F on each Grassmannian of planes in P. If F has k-dimensional support, we show that the Chow form of F is the determinant of the Chow complex of F on the Grassmannian of planes of codimension k + 1. The Beilinson monad of F [Beilinson 1978] is the Chow complex of F on the Grassmannian of 0-planes (that is, on P itself.) In particular, we are able to give explicit determinantal and Pfaffian formulas for resultants in some cases where no polynomial formulas were known. For example, the Horrocks-Mumford bundle gives rise to polynomial formulas for the resultant of five homogeneous forms of degrees 4, 6 or 8 in five variables. The easiest of our new formulas to write down is for the resultant of 3 quadratic forms in three variables, the Chow form of the Veronese surface in P. Using the tangent bundle of P, conclude that it can be written in “Bezout form” (described below) as the Pfaffian of the matrix  0 [245] [345] [135] [045] [035] [145] [235]

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3fac78cc03f8697b02be222379701de77547516e

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Citations in the corpus

Pfaffian quartic surfaces and representations of Clifford algebras (2011, background): “2. The importance of this notion stems from the fact that the Beilinson monad of E (viewed as a coherent sheaf on Pn) reduces to a two-term complex if and only if E is weakly Ulrich (see Section 2 of [ESW]). Moreover, the sole nonzero morphism in this two-term complex is a matrix of linear forms if ”
An introduction to Ulrich bundles (2016, methodology): “2 ([11]) Let X ⊂ P be a smooth variety of dimension n, carrying an Ulrich bundle of rank r .”
n-dimensional Fano varieties of wild representation type (2010, methodology): “A complete account is provided in [12].”
CM Stability and the Generalized Futaki Invariant I (2006, background): “As we will recall in what follows Grothendieck emphasized that one really takes the Chow form of coherent sheaf on P N of dimension n (see [ 11 ] ) .”
ACM bundles on cubic surfaces (2008, background): “In [12], Eisenbud and Schreyer proved the existence of Ulrich bundles on any algebraic curve, on Veronese varieties and of rank two Ulrich sheaves on Del Pezzo surfaces.”
Boij-Söderberg theory: Introduction and survey (2011, background): “f Subsubsection 3.1.7. Instead of Betti diagrams one may consider cohomology tables arising from other gradings. Eisenbud and Schreyer in [12] consider vector bundles Fon P1×P1. The cohomology groups (16) HiF(a,b), i = 0,1,2, (a,b) ∈Z2 give a cohomology table in ⊕ (a,b)∈Z2Q 3. One gets a positive ra”
The representation type of Segre varieties (2012, background): “In fact, few examples of varieties supporting Ulrich sheaves are known, although in [9] it has been conjectured that any variety has an Ulrich sheaf.”
Non-commutative desingularization of determinantal varieties I (2009, neutral): “(The interested reader may wish to compare the Appendix by Weyman to [10], which, by different methods, computes as a special case Ext( ∧p Ω, ∧q Ω) for all i > 0.”
Non-Commutative Crepant Resolutions: Scenes from Categorical Geometry (2011, methodology): “gy Hi(P,HomO P(Ω b−1(b),Ωa−1(a))(c)); since p isﬂat,thiswillcomputeExti OZ (Mb,Ma)aswell. In[BLV10b]wegaveacharacteristicfree proof of this vanishing, and another appears in the appendix by Weyman to [ESW03]. In characteristic 0, one can compute the cohomology with Bott vanishing [Wey03, Chapter 4].”
THE GEOMETRY OF ULRICH BUNDLES ON DEL PEZZO SURFACES (2011, methodology): “ by a homogeneous form f(x0,···,xn), Ulrich bundles on X correspond to linear determinantal descriptions of powers of f (e.g. [Bea]). This has been generalized to the case of arbitrary codimension in [ESW], where Ulrich bundles are shown to correspond to linear determinantal descriptions of powers o”