Unirationality of Cubic Hypersurfaces

Fano Varieties of Low-degree Smooth Hypersurfaces and Unirationality

Degree of Unirationality for Del Pezzo Surfaces over Finite Fields

We address the question of the degree of unirational parameterizations of degree four and degree three del Pezzo surfaces. Specifically we show that degree four del Pezzo surfaces over finite fields admit degree two parameterizations and minimal cubic surfaces admit parameterizations of degree 6. It is an open question whether or not minimal cubic surfaces over finite fields can admit degree 3 ...

Weak Approximation for Cubic Hypersurfaces

We prove weak approximation for smooth cubic hypersurfaces of dimension at least 2 defined over the function field of a complex curve.

RATIONAL POINTS ON CUBIC HYPERSURFACES OVER Fq(t)

The Hasse principle and weak approximation is established for non-singular cubic hypersurfaces X over the function field Fq(t), provided that char(Fq) > 3 and X has dimension at least 6.

Looking for Rational Curves on Cubic Hypersurfaces

The aim of these lectures is to study rational points and rational curves on varieties, mainly over finite fields Fq. We concentrate on hypersurfaces Xn of degree ≤ n+ 1 in Pn+1, especially on cubic hypersurfaces. The theorem of Chevalley–Warning (cf. Esnault’s lectures) guarantees rational points on low degree hypersurfaces over finite fields. That is, if X ⊂ Pn+1 is a hypersurface of degree ≤...