Sublines of Prime Order Contained in the Set of Internal Points of a Conic

AbstractIn [2] it was shown that if q ≥ 4n2−8n+2 then there are no subplanes of order q contained in the set of internal points of a conic in PG(2,qn), q odd, n≥ 3. In this article we improve this bound in the case where q is prime to $q > 2n^2-(4-2\sqrt{3})n+(3-2\sqrt{3})$, and prove a stronger theorem by considering sublines instead of subplanes. We also explain how one can apply this result to flocks of a quadratic cone in PG(3,qn), ovoids of Q(4,qn), rank two commutative semifields, and eggs in PG(4n−1,q).

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