Related papers: Halphen conditions and postulation of nodes

Motivated by the weighted Bounded Negativity Conjecture, we prove that all but finitely many reduced and irreducible curves $C$ on the blow-up of $\mathbb{P}^2$ at $n$ points satisfy the inequality $C^2 \ge \min \{-\frac{1}{12} n (C.L +27),…

Let $X$ be a projective variety and let $C$ be a rational normal curve on $X$. We compute the normal bundle of $C$ in a general complete intersection of hypersurfaces of sufficiently large degree in $X$. As a result, we establish the…

Let X be a complex manifold and c a simple closed curve in X. We address the question: What conditions on c ensure the existence of a 1-dimensional complex subvariety V with boundary c in X. When X = C^n, an answer to this question involves…

We give an explicit construction of a closed curve with constant torsion and everywhere positive curvature. We also discuss the restrictions on closed curves of constant torsion when they are constrained to lie on convex surfaces.

Fix integers $r\geq 4$ and $i\geq 2$. Let $C$ be a non-degenerate, reduced and irreducible complex projective curve in $\mathbb P^r$, of degree $d$, not contained in a hypersurface of degree $\leq i$. Let $p_a(C)$ be the arithmetic genus of…

Suppose $Y$ is a smooth variety equipped with a top form. We prove a simple theorem giving a sharp lower bound on the geometric genus of a family of subvarieties of $Y$, in terms of the dimension of this family. Two elementary applications…

A projectability result is proved for surfaces of prescribed mean curvature (shortly called $H$-surfaces) spanned in a partially free boundary configuration. Hereby, the $H$-surface is allowed to meet the support surface along its free…

Given a connected smooth projective surface X over the complex numbers, together with a simple normal crossings divisor D on it, we study finite normal covers Y of X that are unramified outside D. Given moreover a fibration of X onto a…

In this work we study the connection between the existence of finite dihedral covers of the projective plane ramified along an algebraic curve C, infinite dihedral covers, and pencils of curves containing C.

We prove the regularity conjecture, namely Eisenbud-Goto conjecture, for a normal surface with rational, Gorenstein elliptic and log canonical singularities. Along the way, we bound the regularity for a dimension zero scheme by its Loewy…

We study the intersection theory of punctured pseudoholomorphic curves in $4$-dimensional symplectic cobordisms. We first study the local intersection properties of such curves at the punctures. We then use this to develop topological…

We investigate the projective normality of smooth, linearly normal surfaces of degree 9. All non projectively normal surfaces which are not scrolls over a curve are classified. Results on the projective normality of surface scrolls are also…

We characterize contractible curves on proper normal algebraic surfaces in terms of complementary Weil divisors. Using this we generalize the classical criteria of Castelnuovo and Artin. As application we derive a finiteness result on…

In part I we reduced the arithmetic (characteristic zero) version of the P \not \subseteq NP conjecture to the problem of showing that a variety associated with the complexity class NP cannot be embedded in the variety associated the…

We obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we give applications in the studies of the…

Immersions of graphs to the projective plane are studied. A classification of immersions up to regular homotopy is given. A complete invariant of immersions up to regular homotopy is constructed. Equivalence classes are described.

We introduce and study embeddings of graphs in finite projective planes, and present related results for some families of graphs including complete graphs and complete bipartite graphs. We also make connections between embeddings of graphs…

In this note we study the connection between the existence of a projective reconstruction and the existence of a fundamental matrix satisfying the epipolar constraints.

We prove that circle graphs (intersection graphs of circle chords) can be embedded as intersection graphs of rays in the plane with polynomial-size bit complexity. We use this embedding to show that the global curve simplification problem…

Let (C, p_1, p_2, \ldots, p_n) be a general marked curve of genus g, and q_1, q_2, ..., q_n \in P^r be a general collection of points. We determine when there exists a nondegenerate degree d map f : C \to P^r so that f(p_i) = q_i for all i.…