{"ID":22994688,"CreatedAt":"2026-09-17T04:01:24.020413181Z","UpdatedAt":"2026-09-17T04:01:24.020413181Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.18921","arxiv_id":"2609.18921","title":"Bertini's theorem for $F$-rationality is false","abstract":"Let $k=\\overline{\\mathbb{F}_2}$. We construct a nine-dimensional $F$-rational affine variety $X$ admitting a locally closed embedding $X\\hookrightarrow \\mathbb{P}_k^{19}$ together with a dense open set $U\\subseteq (\\mathbb{P}^{19})^\\vee$ such that $X\\cap H$ is not $F$-rational (or even $F$-injective) for all $H\\in U$. We also construct a nine-dimensional projective $F$-rational variety $\\mathfrak{X}$ and a closed embedding $\\mathfrak{X}\\hookrightarrow\\mathbb{P}^{29}_k$ under which the same conclusion holds.","short_abstract":"Let $k=\\overline{\\mathbb{F}_2}$. We construct a nine-dimensional $F$-rational affine variety $X$ admitting a locally closed embedding $X\\hookrightarrow \\mathbb{P}_k^{19}$ together with a dense open set $U\\subseteq (\\mathbb{P}^{19})^\\vee$ such that $X\\cap H$ is not $F$-rational (or even $F$-injective) for all $H\\in U$....","url_abs":"https://arxiv.org/abs/2609.18921","url_pdf":"https://arxiv.org/pdf/2609.18921v1","authors":"[\"Thomas Polstra\",\"Austyn Simpson\"]","published":"2026-09-16T16:58:16Z","proceeding":"math.AG","tasks":"[\"math.AG\",\"math.AC\"]","methods":"[]","has_code":false}