Related papers: How curved is a random complex curve?

We prove that the expected area of the amoeba of a complex plane curve of degree $d$ is less than $\displaystyle{3\ln(d)^2/2+9\ln(d)+9}$ and once rescaled by $\ln(d)^2$, is asymptotically bounded from below by $3/4$. In order to get this…

"Most" hypersurfaces in projective space are irreducible, and rather precise estimates are known for the probability that a random hypersurface over a finite field is reducible. This paper considers the parametrization of space curves by…

In this paper we collect the main properties of free curves in the complex projective plane and a lot of conjectures and open problems, both old and new. In the quest to understand the mystery of free curves, many tools were developed and…

In the article, we exhibit a series of new examples of rigid plane curves, that is, curves, whose collection of singularities determines them almost uniquely up to a projective transformation of the plane.

We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties…

Interesting data often concentrate on low dimensional smooth manifolds inside a high dimensional ambient space. Random projections are a simple, powerful tool for dimensionality reduction of such data. Previous works have studied bounds on…

In this paper, we provide a lower bound for the Cheeger constant and the spectral gap for random complex curves in $\C P^2$. The complex curve is endowed with the restriction of the ambient Fubini-Study metric, and the probability measure…

We propose a computation of curvature of arbitrary two-dimensional surfaces of three-dimensional objects, which is a contribution to discrete gravity with potential applications in network geometry. We begin by linking each point of the…

The `linear orbit' of a plane curve of degree d is its orbit in the projective space of dimension d(d+3)/2 parametrizing such curves under the natural action of PGL(3). In this paper we compute the degree of the closure of the linear orbits…

Let $C$ be an irreducible projective plane curve in the complex projective space ${\mathbb{P}}^2$. The classification of such curves, up to the action of the automorphism group $PGL(3,{\mathbb{C}})$ on ${\mathbb{P}}^2$, is a very difficult…

A marginally outer trapped hypersurface is a generalization of minimal hypersurfaces originated from general relativity. We show a curvature estimate for stable marginally outer trapped hypersurfaces up to the free boundary satisfying a…

We establish a curvature estimate for classical minimal surfaces with total boundary curvature less than 4\pi. The main application is a bound on the genus of these surfaces depending solely on the geometry of the boundary curve. We also…

For any integer $n\geq 2$, we prove that for any large enough integer $d$, with large probability the injectivity radius of a random degree $d$ complex hypersurface in $\C P^n$ is larger than $d^{-\frac{1}2(3n+2)}$. Here the hypersurface is…

Curvature is a fundamental geometric characteristic of smooth spaces. In recent years different notions of curvature have been developed for combinatorial discrete objects such as graphs. However, the connections between such discrete…

We study curve singularities in a smooth surface relative to a smooth boundary curve. We consider the semiuniversal deformations and equisingular deformations of curves with a fixed local intersection number $w$ with the boundary, and prove…

In this paper, we classify the class of constant weighted curvature curves in the plane with a log-linear density, or in other words, classify all traveling curved fronts with a constant forcing term in $\Bbb R^2.$ The classification gives…

We give a sharp lower bound on the area of the domain enclosed by an embedded curve lying on a two-dimensional sphere, provided that geodesic curvature of this curve is bounded from below. Furthermore, we prove some dual inequalities for…

We study ramified covers of the projective plane. Given a smooth projective surface S and a generic enough projection of S to the projective plane, we get a cover of the plane ramified over a plane curve. The branch curve is usually…

The `linear orbit' of a plane curve of degree d is its orbit in P^{d(d+3)/2} under the natural action of PGL(3). We classify curves with positive dimensional stabilizer, and we compute the degree of the closure of the linear orbits of…

We estimate from below the expected Betti numbers of real hypersurfaces taken at random in a smooth real projective n-dimensional manifold. These random hypersurfaces are chosen in the linear system of a large d-th power of a real ample…