Related papers: On determinantal equations for curves and Frobeniu…

While self-similar sets have no tangents at any single point, self-affine curves can be smooth. We consider plane self-affine curves without double points and with two pieces. There is an open subset of parameter space for which the curve…

A determination of the fixed components, base points and irregularity is made for arbitrary numerically effective divisors on any smooth projective rational surface having an effective anticanonical divisor. All of the results are proven…

This is a survey on the Fano schemes of linear spaces, conics, rational curves, and curves of higher genera in smooth projective hypersurfaces, complete intersections, Fano threefolds, etc.

Let X be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli scheme of rank-2 bundles. We show that up to isomorphism, there is only one (up to…

A Riemannian metric is called Hessian if, locally, it can be written as the Hessian of a function called the Hessian potential. A (flat) Manin-Frobenius manifold is a flat Riemannian manifold furnished with a commutative and associative…

Given a hypersurface defined by $f$ in a smooth complex algebraic variety $X$, and a point $P$ on this hypersurface, we consider the invariant $\beta_P(f)$ given by the log canonical threshold at $P$ of ${\mathfrak m}_P\cdot J_f$, where…

In this paper, we explicitly prove that statistical manifolds, related to exponential families and with flat structure connection have a Frobenius manifold structure. This latter object, at the interplay of beautiful interactions between…

In this paper, we extend our result in [3] to hypersurfaces of any smooth projective variety $Y$. Precisely we let $X_0$ be a generic hypersurface of $Y$ and $c_0:\mathbf P^1\to X_0$ be a generic birational morphism to its image, i.e.…

Consider the scheme parametrizing non-constant morphisms from a fixed projective curve to a projective surface. There is a rational map between this scheme and the Chow variety of $1$-cycles on the surface. We prove that, if the curve is…

By a theorem of Wahl, the canonically embedded curves which are hyperplane section of K3 surfaces are distinguished by the non-surjectivity of their Wahl map. In this paper we address the problem of distinguishing hyperplane sections of…

We establish a type of the Picard's theorem for entire curves in $P^n(\mathbb C)$ whose spherical derivative vanishes on the inverse images of hypersurface targets. Then, as a corollary, we prove that there is an union $D$ of finite number…

The main goal of this work is to prove that every entire curve in a smooth hypersurface of degree greater than or equal to 97 in the complex projective space of dimension 4 must satisfy an algebraic differential equation of order 3. A…

I give a theory of Moebius-flat hypersurfaces in n-dimensional projective space, analogous to that in conformal geometry. This unifies the classes of hypersurfaces with flat induced conformal structure (n > 3) and a classically studied…

If $X = V(f) \subset \mathbb P^N$ is a reduced complex hypersurface, the hessian of $f$ (or by abusing the terminology the hessian of $X$) is the determinant of the matrix of the second derivatives of the form $f$, that is the determinant…

We show that every automorphism of the Hilbert scheme of $n$ points on a weak Fano or general type surface is natural, i.e. induced by an automorphism of the surface, unless the surface is a product of curves and $n=2$. In the exceptional…

A smooth ruled surface in 4-space has only parabolic points or inflection points of real type. We show, by means of contact with transverse planes, that at a parabolic point, there exist two tangent directions determining two planes along…

The first main result of this note, Theorem 1.2, establishes the determinantal identities (7) and (8) for the expectation, under a determinantal point process governed by an integrable projection kernel, of scaling limits of characteristic…

We initiate the study of a class of real plane algebraic curves which we call expressive. These are the curves whose defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of a…

In this article we extend the notion of determinantal representation of hypersurfaces to the determinantal representation of sections of the determinant line bundle of a vector bundle. We give several examples, and prove some necessary…

We prove the Integral Hodge Conjecture for curve classes on smooth varieties of dimension at least three with nef anticanonical divisor constructed as a complete intersection of ample hypersurfaces in a smooth toric variety. In particular,…