Integral Points on Cubic Hypersurfaces

Counting Rational Points on Cubic Hypersurfaces

Let X ⊂ P be a geometrically integral cubic hypersurface defined over Q, with singular locus of dimension 6 dimX − 4. Then the main result in this paper is a proof of the fact that X(Q) contains Oε,X(B ) points of height at most B.

N ov 2 00 6 INTEGRAL POINTS ON CUBIC HYPERSURFACES

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 ≤...

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.

Star points on smooth hypersurfaces

— A point P on a smooth hypersurface X of degree d in PN is called a star point if and only if the intersection of X with the embedded tangent space TP (X) is a cone with vertex P . This notion is a generalization of total inflection points on plane curves and Eckardt points on smooth cubic surfaces in P3. We generalize results on the configuration space of total inflection points on plane curv...