Related papers: Projective structures on curves and conformal geom…
We generalize results by Wakabayashi and Orevkov about rational cuspidal curves on the projective plane to that on $\mathbb{Q}$-homology projective planes. It turns out that the result is exactly the same as the projective plane case under…
Every compact Riemann surface $X$ admits a natural projective structure $p_u$ as a consequence of the uniformization theorem. In this work we describe the construction of another natural projective structure on $X$, namely the Hodge…
In this article, we study isomorphisms between complements of irreducible curves in the projective plane $\mathbb{P}^2$, over an arbitrary algebraically closed field. Of particular interest are rational unicuspidal curves. We prove that if…
We study complex analytic projective connections on surfaces in projective n-spaces in terms of the "second" neighborhood of the surface in the ambient space, and in terms of the osculating behavior of the integral curves. We also…
Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard…
In this paper we describe projective curves and surfaces such that almost all their hyperplane sections are projectively equivalent. Our description is complete for curves and close to being complete for smooth surfaces. In the appendix we…
The purpose of this article is to give an interpretation of real projective structures and associated cohomology classes in terms of connections, sections, etc. satisfying elliptic partial differential equations in the spirit of Hodge…
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…
The purpose of this paper is to present projective geometry in a synthetic, visual and intuitive style through the central notion of harmonicity which leads to harmonic curves. This presentation includes new results, unpublished proofs of…
We investigate the duality between local (complex analytic) projective structures on surfaces and two dimensional (complex analytic) neighborhoods of rational curves having self-intersection +1. We study the analytic classification,…
In this paper we consider Yamabe type problem for higher order curvatures on manifolds with totally geodesic boundaries. We prove local gradient and second derivative estimates for solutions to the fully nonlinear elliptic equations…
It is shown using a space-time curvature classification and decomposition that for certain holonomy types of a space-time, proper projective vector fields cannot exist. Existence is confirmed, by example, for the remaining holonomy types.…
Obtaining complete information about the shape of an object by looking at it from a single direction is impossible in general. In this paper, we theoretically study obtaining differential geometric information of an object from orthogonal…
We continue our previous work studying critical exponent semilinear elliptic (and subelliptic) problems which generalize the classical Yamabe problem. In [3] the focus was on metric-measure spaces with an `almost smooth' structure, with…
The complex projective structures considered is this article are compact curves locally modeled on $\mathbb{CP}^1$. To such a geometric object, modulo marked isomorphism, the monodromy map associates an algebraic one: a representation of…
Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to…
This paper is devoted to the investigation of selected situations when the computation of projective (and other) equivalences of algebraic varieties can be efficiently solved with the help of finding projective equivalences of finite sets…
We provide supplements and open problems related to structure theorems for maximal rationally connected fibrations of certain positively curved projective varieties, including smooth projective varieties with semi-positive holomorphic…
We study the Yamabe problem on open manifolds of bounded geometry and show that under suitable assumptions there exist Yamabe metrics, i.e. conformal metrics of constant scalar curvature. For that, we use weighted Sobolev embeddings.
A systematic review of the various topologies that can be defined on the projective Hilbert space P(H), i.e., on the set of the pure quantum states, is presented. It is shown that P(H) carries a natural topology as well as a natural…