In a projective plane πq of order q, a non-empty point set St is a t-semiarc if the number of tangent lines to St at each of its points is t. If St is a t-semiarc in πq, t < q, then each line intersects St in at most q + 1-t points. Dover proved that semiovals (semiarcs with t = 1) containing q collinear points exist in πq only if q ≤ 3. We show that if t > 1, then t-semiarcs with q + 1-t collinear points exist only if t ≥ √q-1. In PG(2; q) we prove the lower bound t ≥ (q-1)/2, with equality only if St is a blocking set of Rédei type of size 3(q + 1)/2. We call the symmetric difference of two lines, with t further points removed from each line, a Vt-configuration. We give conditions ensuring a t-semiarc to contain a Vt-configuration and give the complete characterization of such t-semiarcs in PG(2; q).

Semiarcs with long secants / Csajbok, Bence. - In: ELECTRONIC JOURNAL OF COMBINATORICS. - ISSN 1077-8926. - ELETTRONICO. - 21:1(2014). [10.37236/3771]

Semiarcs with long secants

Bence Csajbok
2014-01-01

Abstract

In a projective plane πq of order q, a non-empty point set St is a t-semiarc if the number of tangent lines to St at each of its points is t. If St is a t-semiarc in πq, t < q, then each line intersects St in at most q + 1-t points. Dover proved that semiovals (semiarcs with t = 1) containing q collinear points exist in πq only if q ≤ 3. We show that if t > 1, then t-semiarcs with q + 1-t collinear points exist only if t ≥ √q-1. In PG(2; q) we prove the lower bound t ≥ (q-1)/2, with equality only if St is a blocking set of Rédei type of size 3(q + 1)/2. We call the symmetric difference of two lines, with t further points removed from each line, a Vt-configuration. We give conditions ensuring a t-semiarc to contain a Vt-configuration and give the complete characterization of such t-semiarcs in PG(2; q).
2014
Semiarcs with long secants / Csajbok, Bence. - In: ELECTRONIC JOURNAL OF COMBINATORICS. - ISSN 1077-8926. - ELETTRONICO. - 21:1(2014). [10.37236/3771]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11589/234054
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