Summary: UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLV
2007
REMARKS ON THE CAYLEY­VAN DER WAERDEN­CHOW
FORM
by R¨udiger Achilles and J¨urgen St¨uckrad
Abstract. It is known that a variety in projective space is uniquely de-
termined by its Cayley­van der Waerden­Chow form. An algebraic formu-
lation and a proof (for an arbitrary base field) of this classical result are
given in view of applications to the St¨uckrad­Vogel intersection cycle.
1. Introduction. It is well-known that, given a k-dimensional projective
variety X Pn
K, all (n - k - 1)-dimensional projective subspaces meeting X
form a hypersurface in the Grassmannian of (n-k -1)-dimensional projective
subspaces in Pn from which X can be recovered. The homogeneous form in
the Pl¨ucker coordinates defining this hypersurface is known as the (Cayley­van
der Waerden­) Chow form of X. It was introduced by Cayley [2], [3] and later
generalized by Chow and van der Waerden [4]. Since then there has appeared
a vast literature on the subject, see for example [5], [7], and the references
given there.
In this note we present some results on generic hyperplane sections of affine

Source: Achilles, Rüdiger - Dipartimento di Matematica, Università di Bologna

Collections: Mathematics