Pencils of Hyperconics in Projective Planes of Characteristic Two

Let F be any field, finite or infinite, of characteristic 2. Put =PG(2,F). Let H ₁,H ₂ be hyperconics in . In this note we study the intersectionH ₁ H ₂. In particular we obtain canonical forms for H ₁,H ₂ in the cases where |H ₁ H ₂|=4,5,6. One interesting consequence is that the case |H ₁ H ₂|=6 can only occur if F contains a subfield of order 4. Related results concerning pencils of hyperconics are presented in Theorems 6 through 9. This work also leads to an extension to general fields of characteristic 2 of the well-known even intersection property for hyperovals in PG(2,4) which is pursued elsewhere (2]).