Third Year Major Oral: Rational Curves on Algebraic Varieties in Positive Characteristic

Abstract: Rational curves on algebraic varieties control their geometry in very significant ways. One important notion involving rational curves on a variety is that of rational connectedness: when the base field is uncountable, we say that a variety X is rationally connected if there is a rational curve passing through any two general points. Over characteristic zero, all Fano varieties are known to be rationally connected. However, in positive characteristic, rational connectedness does not seem to be the right notion to study. Instead, we study what is called separable rational connectedness: when the variety is smooth, this is equivalent to the existence of a very free rational curve on the variety. Whether an arbitrary smooth Fano hypersurface is separably rationally connected is still an open question. In 2022,  Starr-Tian-Zong answered the question affirmatively for large characteristic by proving the existence of free lines on smooth Fano hypersurfaces. The answer remains unknown when the degree is bigger than the characteristic. In this talk, I am going to introduce the necessary background and go over the existing results around that question. Finally, I will talk about our progress in the case of smooth Fano hypersurfaces whose degree is one more than the characteristic of the base field.

Advisor: Roya Beheshti