Related papers: On Algebraic Hyperbolicity of Log Varieties

In this paper we show the existence of a closed, embedded $\lambda$-hypersurfaces $\Sigma \subset \mathbb{R}^{2n}$. The hypersurface is diffeomorhic to $\mathbb{S}^{n-1} \times \mathbb{S}^{n-1} \times \mathbb{S}^1$ and exhibits $SO(n)…

In this paper, we prove there exist at least $[\frac{n+1}{2}]+1$ geometrically distinct closed characteristics on every compact convex hypersurface $\Sg$ in $\R^{2n}$. Moreover, there exist at least $[\frac{n}{2}]+1$ geometrically distinct…

Given a singular hypersurface in a regular 2-dimensional scheme essentially of finite type over a field, we construct an embedded resolution of singularities by weighted blow-ups. This differs from our earlier work which required…

A rigid isotopy of real algebraic curves of a certain class is a path in the space of curves of this class. The paper's study completes the rigid isotopic classification of nonsingular real algebraic curves of bidegree (4,3) on a…

Once first answers in any dimension to the Green-Griffiths and Kobayashi conjectures for generic algebraic hypersurfaces $\mathbb{X}^{n-1} \subset \mathbb{P}^n(\mathbb{C})$ have been reached, the principal goal is to decrease (to improve)…

We study two special families of cubic hypersurfaces with vanishing Hessian in $\mathbb{P}^N$, obtaining rational parametrizations and computing their degree in $\mathbb{P}(S_3)$. For $N \leq 6$, these two families exhaust the locus of…

Let S be an arbitrary real surface, with or without boundary, contained in a hypersurface M of the complex euclidean space \C^2, with S and M of class C^{2, a}, where 0 < a < 1. If M is globally minimal, if S is totally real except at…

We prove that if Y is a hypersurface of degree d in P^n with isolated singularities, then the log canonical threshold of (P^n,Y) is at least min{n/d,1}. Moreover, if d is at least n+1, then we have equality if and only if Y is the…

We compute the essential dimension of the functors Forms_{n,d} and Hypersurf_{n, d} of equivalence classes of homogeneous polynomials in n variables and hypersurfaces in P^{n-1}, respectively, over any base field k of characteristic 0. Here…

We show that Lang's hyperbolic and function version conjectures hold for surfaces $S$ of general type having a fibration of general type onto a curve $C$. The notion of multiplicity used is natural, but not classical, which leds to orbifold…

We study the base locus of the higher fundamental forms of a projective toric variety $X$ at a general point. More precisely we consider the closure $X$ of the image of a map $({\mathbb C}^*)^k\to {\mathbb P}^n$, sending $t$ to the vector…

We study the ergodic theory of non-conservative C^1-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show…

Recently, Corvaja and Zannier obtained an extension of the Subspace Theorem with arbitrary homogeneous polynomials of arbitrary degreee instead of linear forms. Their result states that the set of solutions in P^n(K) (K number field) of the…

Let $n\geq 1$ be an integer, $\mathcal L \subset \mathbb{R}^n$ be a compact smooth affine real hypersurface, not necessarily connected. We prove that there exists $c>0$ and $d_0\geq 1$, such that for any $d\geq d_0$, any smooth complex…

We give a simple topological argument to show that the number of solutions of the asymptotic Plateau problem in hyperbolic space is generically unique. In particular, we show that the space of codimension-1 closed submanifolds of sphere at…

We obtain a lower bound for the normalised height of a non-torsion hypersurface $V$ of a C.M. abelian variety $A$ which is a refinement of a precedent result. This lower bound is optimal in terms of the geometric degree of $V$, up to an…

Alex Waldron proved that for sufficiently general degree $d$ hypersurfaces in projective $n$-space, the Fano scheme parameterizing $r$-dimensional linear spaces contained in the hypersurface is nonempty precisely for the degree range $n\geq…

We give a short proof of the systolic inequality for the n-dimensional torus. The proof uses minimal hypersurfaces. It is based on the Schoen-Yau proof that an n-dimensional torus admits no metric of positive scalar curvature.

We show that a wide class of hypersurfaces in all dimensions are not stably rational. Namely, for all d at least about 2n/3, a very general complex hypersurface of degree d in P^{n+1} is not stably rational. The statement generalizes…

We prove a conjecture of Voisin that no two distinct points on a very general hypersurface of degree $2n$ in ${\mathbb P}^n$ are rationally equivalent.