Riv.Mat.Univ.Parma (7) 2 (2003)


On two theorems of Bertini for infinite-dimensional projective spaces

Pages: 1-7
Received: 16 October 2002   
Mathematics Subject Classification (2000): 32K05 - 14N05

Work partially supported by MURST and GNSAGA of INdAM (Italy)

Abstract: Here we prove the following two results. Fix positive integers s, d1, ... , ds and a separable Hilbert space V; then there exist smooth codimension s complete intersection X ⊂ P (V) of s hypersurfaces of degree d1, ... , ds. Fix an integer d ≥ 2 and a subset S of CP1 with at most countable elements; then there exist linearly independent homogeneous degree d polynomials F and G on C(N) such that a hypersurface { λF + μG = 0} of P(C(N)) with (λ; μ) ∈ CP1 is singular if and only if (λ; μ)∈S; we allow the case S=Ø, which is in striking contrast with the corresponding problem in CPn.

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