Speaker: Nour Alnajjarine
Title: Solids and the Veronese variety in PG(5, q), q even
Date/Time: 10 March 2021 / 13:30 - 14:30
Zoom: Meeting ID: https://sabanciuniv.zoom.us/s/91735827843
Abstract:Let W be the 6-dimensional vector space of the 2-forms defined on PG(2, q). Subspaces of PG(W) are called linear systems of conics. In particular, 1-dimensional subspaces are the pencils of conics, 2-dimensional subspaces are the nets of conics and 3-dimensional subspaces are the webs of conics. By observing that linear systems of conics correspond to subspaces of PG(5, q), it follows that classifying these systems up to projective equivalence corresponds to classifying subspaces of PG(5, q) under the induced action of the subgroup K < PGL(6, q) equivalent to PGL(3, q). Recently, Lines and planes meeting the Veronese variety in at least one point were classified by Lavrauw et. al [2, 3] over all fields and finite fields of odd order respectively. Their results proved the existence of 15 such K-orbits, which lead to the existence of 15 webs and rank-one nets of conics up to projective equivalence. Following a similar approach, we were able to classify solids in PG(5, q), q even. Our result completes the partial classification of pencils of conics in PG(2, 2n) provided by Campbell in  and corrects two of his invalid claims.
In this talk, we will explain the 15 K-orbits of solids in PG(5, q), q even, by showing some
of their properties and complete invariants. We end with a table comparing our K-orbits with
the previous incomplete classification in .
 Campbell, A.: Pencils of Conics in the Galois Fields of order 2n. Amer. J. Math, 49, 3: 401-406
 Lavrauw, M., Popiel, T.: The symmetric representation of lines in PG(F3q ⊗F3q). Discrete Math. 343, 4 (2020).
 Lavrauw, M., Popiel, T., Sheekey, J.: Nets of conics of rank one in PG(2, q), q odd. J. Geom. 111, 36 (2020).