Philip P. W. Wong

On embedding a unitary block design as a polar unital and an intrinsic characterization of the classical unital

A necessary and sufficient condition is given for embedding a unital into a projective plane as a polar unital. A strengthened version of the condition is introduced and is shown to be necessary for a classical unital. Using the strengthened condition and results of Wilbrink (1983) and Grundhofer, Stroppel and Van Maldeghem (2013), a new intrinsic characterization of the classical unital is given without assuming the absence of O?Nan configurations. Finally, a unital of even order satisfying the first two intrinsic characterization conditions of Wilbrink is shown to satisfy the strengthened condition by an elementary (combinatorial-geometric) proof and without invoking deep results from group theory.

On the structure of the Figueroa unital and the existence of O’Nan configurations

Abstract The finite Figueroa planes are non-Desarguesian projective planes of order q 3 for all prime powers q > 2 , constructed algebraically in 1982 by Figueroa, and Hering and Schaeffer, and synthetically in 1986 by Grundhofer. All Figueroa planes of finite square order are shown to possess a unitary polarity by de Resmini and Hamilton in 1998, and hence admit unitals. Hui and Wong (2012) have shown that these polar unitals do not satisfy a necessary condition, introduced by Wilbrink in 1983, for a unital to be classical, and hence they are not classical. In this article we introduce and make use of a new alternative synthetic description of the Figueroa plane and unital to demonstrate the existence of O’Nan configurations, thus providing support to Piper’s conjecture (1981).

A geometric proof of a theorem on antiregularity of generalized quadrangles

A geometric proof is given in terms of Laguerre geometry of the theorem of Bagchi, Brouwer and Wilbrink, which states that if a generalized quadrangle of order s > 1 has an antiregular point then all of its points are antiregular.

The concepts of general position and a second main theorem for non-linear divisors

The recent works of Evertse–Ferretti (Evertse and Ferretti, A generalization of the subspace theorem with polynomials of high degree, Dev. Math. (2008), pp.175–198) and Corvaja–Zannier (Corvaja and Zannier, On a general Theus equation, Ann. Math. 160 (2004), pp. 705–726; Corvaja and Zannier, On integral points on surfaces, Amer. J. Math. 126 (2004), pp. 1033–1055) in diophantine approximations (Ru, A defect relation for holomorphic curves intersecting hypersurfaces, Amer. J. Math. 126 (2004), pp. 215–266; An, A defect relation for non-Archimedean analytic curves in arbitrary projective varieties, Proc. Amer. Math. Soc. 135 (2007), pp. 1255–1261) in complex and p-adic Nevanlinna theory extend the classical subspace theorem and the classical second main theorem to the case of non-linear divisors in general position. These had been long standing problems in diophantine approximation and Nevanlinna theory. However, their results when specialized to the case of hyperplanes are weaker than the classical results. In this article, we refine the concept of general position to the concepts of p-jet general position. These concepts of general position involve jets of order p and coincide with the usual concept of general position for hyperplanes, but are different for hypersurfaces of higher degrees. With the assumption that the hypersurfaces are in n-jet general position, a second main theorem, with ramification term, for non-linear divisors and d-non-degenerate map f : ℂ → ℙ n is obtained. The result when specialized to hyperplanes is precisely the classical result of Ahlfors (The theory of meromorphic curves, Acta Soc. Sci. Fenn. 3 (1941), pp. 1–31) (see also Cartan, Sur les zeros des combinations lineares de p fonctions holomorphes donees, Mathematica 7 (1933), pp. 5–31; Stoll, About the value distribution of holomorphic maps into the projective space, Acta Math. 123 (1969), pp. 83–114; Cowen and Griffiths, Holomorphic curves and metrics of non-negative curvature, J. Anal. Math. 29 (1976), pp. 93–153; for small functions, see Yamanoi, The second main theorem for small functions and related problems, Acta Math. 192 (2004), pp. 225–294). In fact the proof is a modification of Ahlfors proof. There are a number of variations of Ahlfors proof, we choose the ‘approximate negatively curved’ approach used in Cowen and Griffiths (Holomorphic curves and metrics of non-negative curvature, J. Anal. Math. 29 (1976), pp. 93–153), Wong (Defect relation for meromorphic maps on parabolic manifolds and Kobayashi metrics on ℙn omitting hyperplanes, Ph.D. thesis, University of Notre Dame (1976)), Wong (Holomorphic curves in spaces of constant curvature, in Complex Geometry (Proceedings of the Osaka Conference, 1990), Lecture Notes in Pure and Applied Mathematics, Vol. 143, Marcel Dekker, New York, Basel, Hong Kong, 1993, pp. 201–223), Wong (On the second main theorem of Nevanlinna theory, Amer. J. of Math. 111 (1989), pp. 549–583) and Cowen (The Kobayashi metric on P n ∖ (2n + 1) hyperplanes, in Value Distribution Theory, Part A, R.O. Kujala and A.L. Vitter III, eds., Marcel Dekker, New York, 1974, pp. 205–223).