On incidence structures of nonsingular points and hyperbolic lines of ovoids in finite orthogonal spaces

Abstract

The revisions were made according to the referees' suggestions as mentioned in the bold and italic statements below.Reviewer #1: You may have missed the recent paper A classification of transitive ovoids, spreads and m-systems of polar spaces, Forum Mathematicum 21(2009), 181-216 which classifies primitive ovoids in all classical polar spaces provided they do not have cardinality a prime number. I recommend that you add it to your list of references, and refer to it in Section 1 (after discussing Kleidman's result) and/or in Section 5, when referring to your own result about primitive ovoids from [15].Added to the list of references. Refered to it in Section 1 and Section 5.Typos : voids for ovoids (page 13, line 9), bipartitie for bipartite (page 6, line 21); there is no Tits ovoid over the field of order 27 (page 13, line 28), but they should have occurred two sentences earlier in the list of known ovoids. Page 13, line 27, you need two definite articles added : For q=25 the only... For q=27 the only... Typos were corrected. Reviewer 2The authors introduce new invariants of ovoids in finite orthogonal spaces. This paper is suitable for publication in DCC provided the following changes are made: p3 l 49 reference to [19] should be replaced by: Charnes and Dempwolff [CD7], and this reference should be included in the bibliography. It should also be mentioned that fingerprints give an effective method of calculating the automorphism groups of ovoids. The following typos were corrected. Abstract. We study the point-line incidence structures of nonsingular points and hyperbolic secant lines associated with ovoids in finite orthogonal spaces. We show that these incidence structures frequently produce partial linear spaces and the parameters of the bipartite graphs (called ovoidal graphs) associated with these structures produce simple and effective isomorphism invariants to distinguish non-isomorphic ovoids. We prove explicit formulas for these isomorphism invariants for a number of infinite families of 2-transitive ovoids. Abstract. We study the point-line incidence structures of nonsingular points and hyperbolic secant lines associated with ovoids in finite orthogonal spaces. We show that these incidence structures frequently produce partial linear spaces and the parameters of the bipartite graphs (called ovoidal graphs) associated with these structures produce simple and effective isomorphism invariants to distinguish non-isomorphic ovoids. We prove explicit formulas for these isomorphism invariants for a number of infinite families of 2-transitive ovoids.