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.
Bertini theorems for Hilbert-Samuel multiplicity over finite fields
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abstract
Let $X\subseteq \mathbb{P}^n_{\mathbb{F}_q}$ be a reduced, equidimensional, quasiprojective scheme. We prove that there exists a positive-density set of hypersurfaces $H_f$ such that for every closed point $P\in X\cap H_f$, one has $\mathrm{ord}_P(f)=1$ and $e_P(X\cap H_f)=e_P(X)$.
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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.