Let S(Π0,Π1) be the product of the projective spaces Π0 and Π1, i.e. the semilinear space whose point set is the product of the point sets of Π0 and Π1, and whose lines are all products of the kind {P0}×g1 or g0×{P1}, where P0, P1 are points and g0, g1 are lines. An embedding χ : S(Π0,Π1) → Π′ is an injective mapping which maps the lines of S(Π0,Π1) onto (whole) lines of Π′. The classical embed...

Quadratically parametrized maps from a product of real projective spaces to a complex projective space are constructed as the composition of the Segre embedding with a projection. A classification theorem relates equivalence classes of projections to equivalence classes of complex matrix pencils. One low-dimensional case is a family of maps whose images are ruled surfaces in the complex project...

Lehn’s conjecture. The number of (n− 2)-subspaces in P2n−2 which are n-secant to a smooth curve, C ⊂ P2n−2, of genus g and degree d is a classical enumerative calculation [ACGH]. The answer can be expressed in terms of Segre integrals over the symmetric product C [n] of C. Let the line bundle H → C be the degree d restriction of OP2n−2(1). The n-secant problem is solved by the Segre integral, a...

Let χ be a linear morphism of the product of two projective spaces PG(n, F ) and PG(m,F ) into a projective space. Let γ be the Segre embedding of such a product. In this paper we give some sufficient conditions for the existence of an automorphism α of the product space and a linear morphism of projective spaces φ, such that γφ = αχ. A.M.S. classification number: 51M35.

We compute the arithmetic ranks of the defining ideals of homogeneous coordinate rings of certain Segre products arising from elliptic curves. The cohomological dimension of these ideals varies with the characteristic of the field, though the arithmetic rank does not. We also study the related set-theoretic Cohen-Macaulay property for these ideals. In [12] Lyubeznik writes: Part of what makes t...

Let Hn and HN denote the complexifications of Heisenberg hypersurfaces in Cn and CN , respectively. We show that non-degenerate holomorphic Segre mappings from Hn into HN with N ≤ 2n− 2 possess a partial rigidity property. As an application, we prove that the holomorphic Segre non-transversality for a holomorphic Segre map from Hn into HN with N ≤ 2n− 2 propagates along Segre varieties. We also...

The weight of a code is the number of coordinate positions where no codeword is zero. The th minimum weight is the least weight of an -dimensional subcode. Wei and Yang gave a conjecture about the minimum weights for some product codes. In this correspondence, we will find a relation between product codes and the Segre embedding of a pair of projective systems, and we use this to prove the conj...

We prove the GSS conjecture of Garcia, Stillman and Sturmfels, which states that the ideal of the variety of secant lines to a Segre product of projective spaces is generated by 3 × 3 minors of flattenings. We also describe the decomposition of the coordinate ring of this variety as a sum of irreducible representations.

We prove that the stabiliser group GX of the Segre variety product in PG(V ) of three projective lines over a field F has four orbits on singular points of PG(V ), and that GX has five orbits on points of PG(V ) if F is finite.

We give the parameters of any evaluation code on a smooth quadric surface. For hyperbolic quadrics the approach uses elementary results on product codes and the parameters of codes on elliptic quadrics are obtained by detecting a BCH structure on these codes and using the BCH bound. The elliptic quadric is a twist of the surface P ×P and we detect a similar BCH structure on twists of the Segre ...