We discuss the work of Corrado Segre on nodal cubic hypersurfaces with emphasis on the cases of 6-nodal and 10-nodal cubics. In particular we discuss the Fano surface of lines and conic bundle structures on such threefolds. We review some of the modern research in algebraic geometry related to Segre’s work.

We construct a variety of complete plane cubics by a sequence of five blow-ups over P9 . This enables us to translate the problem of computing characteristic numbers for a family of plane cubics into one of computing five Segre classes, and to recover classic enumerative results of Zeuthen and Maillard.

Let Γ′ and Γ be two Grassmannians. The standard embedding φ : Γ′ × Γ→ P is obtained by combining the Plücker and Segre embeddings. Given a further embedding η : Γ′ × Γ→ P′, we find a sufficient condition for the existence of α ∈ Aut(Γ) and of a collineation ψ : P→ P′ such that η = (idΓ′ × α)φψ. A.M.S. classification number: 51M35.

A limiting diagram for the Segre classification in 5-dimensional space-times is obtained, extending a recent work on limits of the energy-momentum tensor in general relativity. Some of Geroch’s results on limits of space-times in general relativity are also extended to the context of five-dimensional Kaluza-Klein space-times. pacs numbers: 04.20.Cv 04.50.+h 04.20.Jb 04.20.-q

Given a flat, projective morphism Y → T from an equidimensional scheme to a nonsingular curve and a subscheme Z of Y , we give conditions under which specialization of the Segre class s(NZY ) of the normal cone of Z in Y implies flatness of the normal cone. We apply this result to study when the relative tangent star cone of a flat family is flat.

We give an almost asymptotically sharp bound for the non-secant defectiveness and identifiability of Segre–Veronese varieties. also provide new examples defective varieties, implement our methods in Magma. Finally, we two applications techniques: classify possibly singular 2-secant toric surfaces study secant Losev–Manin spaces.