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Bertini theorems for hypersurface sections containing a subscheme over finite fields
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We show the existence of a hypersurface that contains a given closed subscheme of a projective space over a finite field and intersects a smooth quasi-projective scheme smoothly, under some condition on the dimension. This generalizes a Bertini theorem by Poonen and is the finite field analogue of a Bertini theorem by Altman and Kleiman. Furthermore, we add the possibility of modifying finitely many local conditions of the hypersurface. We show that the condition on the dimension is fulfilled for schemes with simple normal crossings and give an application to embeddings into smooth schemes.
Forward citations
Cited by 2 Pith papers
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Frobenius-orbit slicing and uniform elimination of positive-dimensional singular loci
Random hypersurface sections over finite fields are shown to avoid positive-dimensional singular loci with probability at least 1 - O((d+1)^r p^{-ceil(d/2)}), proving Poonen's arithmetic Bertini conjecture.
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Bertini theorems for Hilbert-Samuel multiplicity over finite fields
Proves existence of positive-density hypersurfaces over finite fields intersecting a reduced equidimensional quasiprojective scheme X such that multiplicity e_P is preserved at all closed points P of the intersection.
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