Abstract Let Y be a smooth complete intersection of quadric and cubic in ℙ n {\mathbb{P}^{n}} , with n even. We show that has multiplicative Chow–Künneth decomposition, the sense Shen–Vial. As consequence, Chow ring (powers of) displays K3-like behavior. by-product argument, we also esta...

In this paper we study a double cover ψ : X → V ⊂ P branched over a smooth divisor S ⊂ V such that n ≥ 7, the degree of V is 3 or 4, and S is cut on the hypersurface V by a hypersurface of degree 2(n− deg(V )). The variety X is rationally connected, because it is a smooth Fano variety. We prove that X is birationally superrigid. In particular, the variety X is nonrational.

Let X be a smooth hypersurface of degree n ≥ 3 in P. We prove that the log canonical threshold of H ∈ |−KX| is at least n−1 n . Under the assumption of the Log minimal model program, we also prove that a hyperplane section H of X is a cone in P over a smooth hypersurface of degree n in P if and only if the log canonical threshold of H is n−1 n . Bibliography : 20 titles.

We introduce a Lie algebra of initial terms of logarithmic vector fields along a hypersurface singularity. Extending the formal structure theorem in [GS06, Thm. 5.4], we show that the completely reducible part of its linear projection lifts formally to a linear Lie algebra of logarithmic vector fields. For quasihomogeneous singularities, we prove convergence of this linearization. We relate our...

where (ζ : z : w) are homogeneous coordinates in CP, the space C is given in CP by ζ 6= 0 with z = z/ζ , w = w/ζ , and the hyperplane at infinity by ζ = 0. If the form 〈z, z〉 is non-degenerate, then Q〈·,·〉 is a non-singular Levi non-degenerate hypersurface, and it is well-known that the following continuation phenomena hold for CR-automorphisms of Q〈·,·〉: (i) every local C-smooth CR-automorphis...

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 ≤...