For any nonempty smooth quasiprojective pure positive-dimensional variety X over F_q, sufficiently high-dimensional linearly nondegenerate embeddings exist such that every F_q-rational hyperplane section is singular.
Taylor conditions over finite fields
2 Pith papers cite this work. Polarity classification is still indexing.
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abstract
We extend Poonen's Bertini theorem over finite fields to Taylor conditions arising from locally free quotients of the sheaf of differentials on projective space. This is motivated by a result of Bilu and Howe in the motivic setting that allows for significantly more general Taylor conditions.
fields
math.AG 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
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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Hyperplane anti-Bertini embeddings over finite fields
For any nonempty smooth quasiprojective pure positive-dimensional variety X over F_q, sufficiently high-dimensional linearly nondegenerate embeddings exist such that every F_q-rational hyperplane section is singular.
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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.