Related papers: Halphen conditions and postulation of nodes
Let $P$ be a set of $n$ points in general position in the plane. Let $R$ be a set of points disjoint from $P$ such that for every $x,y \in P$ the line through $x$ and $y$ contains a point in $R$. We show that if $|R| < \frac{3}{2}n$ and $P…
Let $M\subset\mathbb{R}^3$ be a properly embedded, connected, complete surface with boundary a convex planar curve $C$, satisfying an elliptic equation $H=f(H^2-K)$, where $H$ and $K$ are the mean and the Gauss curvature respectively -…
A classical example of Mumford gives a generically non-reduced component of the Hilbert scheme of smooth curves in the projective 3-space such that a general element of the component is contained in a smooth cubic hypersurface in the…
In this note we prove a decomposition related to the affine fundamental group and the projective fundamental group of a line arrangement and a reducible curve with a line component. We give some applications to this result.
We classify 'primitive normal compactifications' of C^2 (i.e. normal analytic surfaces containing C^2 for which the curve at infinity is irreducible), compute the moduli space of these surfaces and their groups of auomorphisms. In…
We construct a connection and a curving on a bundle gerbe associated with lifting a structure group of a principal bundle to a central extension. The construction is based on certain structures on the bundle, i.e. connections and…
In this paper we begin to study curves on a weighted projective plane with one trivial weight, ${\mathbb P}(1,m,n)$, by determining the genus of curves of Fermat type. These are curves defined by a ``homogeneous'' polynomial analagous to…
The purpose of this paper is to show that for a complete intersection curve $C$ in projective space (other than a few stated exceptions), any morphism $f: C \to \mathbb{P}^r$ satisfying $\text{deg}\, f^*\mathcal{O}_{\mathbb{P}^r}(1)…
We survey and expand on the work of Segal, Milgram and the author on the topology of spaces of maps of positive genus curves into $n$-th complex projective space, $n\geq 1$ (in both the holomorphic and continuous categories). Both based and…
Recently, there are many developments on the second main theorem for holomorphic curves into algebraic varieties intersecting divisors in general position or subgeneral position. In this paper, we refine the concept of subgeneral position…
The aim of this paper and its sequel is to introduce and classify the holonomy algebras of the projective Tractor connection. After a brief historical background, this paper presents and analyses the projective Cartan and Tractor…
We provide the full classification of algebraic embeddings of $\mathbb{C}^*$ into $\mathbb{C}^2$ satisfying certain regularity condition, which conjecturally holds for all algebraic maps from $\mathbb{C}^*$ into $\mathbb{C}^2$. The…
The authors give a complete classification of projective threefolds admitting a holomorphic normal projective connection. Moreover, they prove a general structure theorem on complex projective manifolds admitting a holomorphic normal…
The relationship between convex geometry and algebraic geometry has deep historical roots, tracing back to classical works in enumerative geometry. In this paper, we continue this theme by studying two interconnected problems regarding…
We propose a linear version of the weighted bounded negativity conjecture. It considers a smooth projective surface $X$ over an algebraically closed field of characteristic zero and predicts the existence of a common lower bound on…
We explore the concept of projections of syzygies and prove two new technical results; we firstly give a precise characterization of syzygy schemes in terms of their projections, secondly, we prove a converse to Aprodu's Projection Theorem.…
We study families V of curves in P^2 of degree d having exactly r singular points of given topological or analytic types. We derive new sufficient conditions for V to be T-smooth (smooth of the expected dimension), respectively to be…
It is a result of Gruson and Peskine that the invariants of a set points in $\ptwo$ in general position are connected. Associated to a space curve there are sequences of invariants which generalize the invariants of points in $\ptwo$. The…
Let $C$ be a smooth irreducible affine curve $C$ over an algebraically closed field of positive characteristic and let $\pi_1(C)$ be its fundamental group. We study various embedding problems for $\pi_1(C)$ and its subgroups.
We propose a geometric correspondence between (a) linearly degenerate systems of conservation laws with rectilinear rarefaction curves and (b) congruences of lines in projective space whose developable surfaces are planar pencils of lines.…