**Speaker:** Güizar Günay

**Title:**On pencils of cubics on the projective line over finite field of characteristic > 3

**Date/Time:** 21 April 2021 / 13:40 - 14:30

**Zoom: Meeting ID**:https://sabanciuniv.zoom.us/j/91735827843?pwd=QlBwc3dUSzlDcHl5OXZUNCs3MWlWZz09

**Passcode: **algebra

**Abstract: **A cubic C in PG(1, q) is the zero locus of a homogenous polynomial f(X_{0}, X_{1}) of degree 3 in Fq[X_{0}, X_{1}]. The cubic forms on PG(1, q) form a four-dimensional vector space W, and subspaces of the projective space PG(W) are called linear systems of cubics. One-dimensional linear systems are called pencils.In this talk, we mention combinatorial invariants of the equivalence classes of pencils of cubics on PG(1, q), for q odd and q not divisible by 3. These equivalence classes are considered as orbits of lines in PG(3, q), under the action of the subgroup G ∼=PGL(2, q) of PGL(4, q) which preserves the twisted cubic C in PG(3, q). In particular we determine the point orbit distributions and plane orbit distributions of all G-orbits of lines which, are contained in an osculating plane of C, have non-empty intersection with C, or are imaginary chords or axes.

**Bio:** -