 
Summary: UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLV
2007
REMARKS ON THE CAYLEYVAN DER WAERDENCHOW
FORM
by Ršudiger Achilles and Jšurgen Stšuckrad
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 StšuckradVogel intersection cycle.
1. Introduction. It is wellknown that, given a kdimensional projective
variety X Pn
K, all (n  k  1)dimensional projective subspaces meeting X
form a hypersurface in the Grassmannian of (nk 1)dimensional projective
subspaces in Pn from which X can be recovered. The homogeneous form in
the Plšucker 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
