Smooth Cubic Hypersurface Of Dimension Research Articles

2-Cycles sur les hypersurfaces cubiques de dimension 5

We study 2-cycles of a smooth cubic hypersurface of dimension 5. We show that the Griffiths group of 2-cycles is trivial and the Abel–Jacobi map induces an isomorphism between the Chow group of algebraically trivial 2-cycles and the intermediate Jacobian.

Hasse principle for three classes of varieties over global function fields

We give a geometric proof that the Hasse principle holds for the following varieties defined over global function fields: smooth quadric hypersurfaces, smooth cubic hypersurfaces of dimension at least 4 in characteristic at least 7, and smooth complete intersections of two quadrics, which are of dimension at least 3, in odd characteristics.

A Mordell–Weil theorem for cubic hypersurfaces of high dimension

Let $X$ be a smooth cubic hypersurface of dimension $n \ge 1$ over the rationals. It is well-known that new rational points may be obtained from old ones by secant and tangent constructions. In view of the Mordell--Weil theorem for $n=1$, Manin (1968) asked if there exists a finite set $S$ from which all other rational points can be thus obtained. We give an affirmative answer for $n \ge 48$, showing in fact that we can take the generating set $S$ to consist of just one point. Our proof makes use of a weak approximation theorem due to Skinner, a theorem of Browning, Dietmann and Heath-Brown on the existence of rational points on the intersection of a quadric and cubic in large dimension, and some elementary ideas from differential geometry, algebraic geometry and numerical analysis.

On the universal $$\mathrm {CH}_0$$ CH 0 group of cubic threefolds in positive characteristic

We adapt for algebraically closed fields k of characteristic >2 two results of Voisin (On the universal $$\text {CH} _0$$ group of cubic hypersurfaces, arXiv:1407.7261 ), on the decomposition of the diagonal of a smooth cubic hypersurface X of dimension 3 over $${\mathbb {C}}$$ , namely: the equivalence between Chow-theoretic and cohomological decompositions of the diagonal of those hypersurfaces and the equivalence between the algebraicity (with $$\mathbb Z_2$$ -coefficients) of the minimal class $$\theta ^4/4!$$ of the intermediate Jacobian J(X) of X and the cohomological (hence Chow-theoretic) decomposition of the diagonal of X. Using the second result, the Tate conjecture for divisors on surfaces defined over finite fields predicts, via a theorem of Schoen (Math Ann 311(3), 493–500, 1998), that every smooth cubic hypersurface of dimension 3 over the algebraic closure of a finite field of characteristic >2 admits a Chow-theoretic decomposition of the diagonal.

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 Curves on Smooth Cubic Hypersurfaces

We prove that the space of rational curves of a fixed degree on any smooth cubic hypersurface of dimension at least four is irreducible and of the expected dimension. Our methods also show that the space of rational curves of a fixed degree on a general hypersurface in of degree 2d ≤ min (n + 4, 2n − 2) and dimension at least three is irreducible and of the expected dimension.

Équivalence rationnelle sur les hypersurfaces cubiques de mauvaise réduction

This article consists of two independent, but related, parts. The first one proves the vanishing of the Chow group of classes of zero-cycles of degree zero modulo rational equivalence for a cubic hypersurface of dimension ⩾10 on a p-adic or C 2 field (and, in fact, the R-triviality of such a hypersurface). This is done without the assumption of good reduction (or even smoothness). The second part goes in the other direction and gives an explicit example of a smooth cubic hypersurface of dimension 3 (necessarily of bad reduction) on a field such as C ( ( ν , t ) ) (or C ( ( ν ) ) ( ( t ) ) ) whose Chow group of classes of zero-cycles of degree zero modulo rational equivalence does not vanish.

Special Cubic Fourfolds

A cubic fourfold is a smooth cubic hypersurface of dimension four; it is special if it contains a surface not homologous to a complete intersection. Special cubic fourfolds form a countably infinite union of irreducible families is isomorphic to the Hilbert scheme of length-two subschemes of an associated K3 surface.