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Blocking sets from a union of plane curves
theorem_3_2: For a reflexive plane curve with geometrically irreducible components defined over an odd finite field, each product conjugacy class occurs among line sections with its group-theoretic density and an error uniform at fixed total degree.
Shamil Asgarli, Dragos Ghioca and Chi Hoi Yip, Blocking sets from a union of plane curves, arXiv:2510.15332v2, 30 September 2026. The first arXiv version appeared in October 2025. The selected edition is v2, held in the PDF, with its arXiv record and source sidecars beside it. The arXiv record fetched on 2026-10-16 states Creative Commons Attribution 4.0.
The paper studies blocking sets that are unions of the rational points of geometrically irreducible nonlinear plane curves. Its Theorem 1.3 excludes a bounded number of fixed-degree curves as the field size grows when the characteristic exceeds their degrees. This restricts the construction class; it does not exclude arbitrary point sets of bounded line intersection. Theorems 1.2 and 1.4 respectively give logarithmic lower bounds for conic families in odd characteristic and logarithmic upper bounds for families of each fixed degree at least three.
The consumed input is Theorem 3.2's Frobenius class count, which the paper attributes to Entin. Its error is uniform at fixed total degree. Section 3 applies it to one full-cycle conjugacy class on each component, obtaining equation (3.1), a positive proportion of lines missing all components. The deductions on exceptional points and curve extraction also use other derangement classes; those deductions belong to the research page, not to the source's theorem label.
Read scope. The abstract, introduction and Section 3 were read from the v2 TeX sidecar. Printed pp. 6–8 were compared with extracted PDF text; Theorem 3.2 and equation (3.1) on printed p. 7 were also checked on the page image. Their statements and the use in (3.1)–(3.2) were followed. Entin's original theorem and Hefez's reflexivity result were not checked, and the other source proofs are not verified here.
Bears on. For Problem 1159, the source supplies the curve part of a possible negative argument in prime planes. Uniform approximation of arbitrary minimal bounded-cap blocking sets by curves remains a separate, unproved premise.