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

**EDOARDO BALLICO**

*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, d*_{1}, ... , *d*_{s}
and a
separable Hilbert space *V*; then there exist smooth codimension *s* complete intersection X ⊂ ** P ** (V)
of s hypersurfaces of degree * d*_{1}, ... , d_{s}. Fix an integer *d* ≥ 2 and a subset * S * of **CP**^{1} 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)
∈
**CP**^{1} 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 ** CP**^{n}.

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