Related papers: Effective algebraic degeneracy

The main result of the paper is the classification of all (nonassociative) algebras of level two, i.e. such algebras that maximal chains of nontrivial degenerations starting at them have length two. During this classification we obtain an…

We show that Kov\'acs' result on the cone of curves of a K3 surface generalizes to any projective irreducible holomorphic symplectic manifold $X$. In particular, we show that if $\rho(X)\geq 3$, the pseudo-effective cone…

We address the following question: Given a polarized toric surface (S,L), and a general integral curve C of geometric genus g in the linear system |L|, do there exist degenerations of C in |L| to general integral curves of smaller geometric…

Let $X\subset \mathbb {P}^r$ be an integral and non-degenerate variety. Set $n:= \dim (X)$. Let $\rho (X)''$ be the maximal integer such that every zero-dimensional scheme $Z\subset X$ smoothable in $X$ is linearly independent. We prove…

Let k be an algebraically closed field complete with respect to a non-Archimedean absolute value of arbitrary characteristic. Let D_1,...,D_n be effective nef divisors intersecting transversally in an n-dimensional nonsingular projective…

In this note, we study various measures of irrationality for hypersurfaces in projective spaces which were recently proposed by Bastianelli, De Poi, Ein, Lazarsfeld and Ullery. In particular, we answer the question raised by Bastianelli…

Given a graph $G$ and a non-negative integer $d$ let $\alpha_d(G)$ be the order of a largest induced $d$-degenerate subgraph of $G$. We prove that for any pair of non-negative integers $k>d$, if $G$ is a $k$-degenerate graph, then…

This note is an appendix to 'Measures of irrationality for hypersurfaces of large degree' by L. Ein, R. Lazarsfeld and B. Ullery. We prove an existence result for families of curves having low gonality, and lying on fundamental loci of…

Let $\mathcal{F}_d(\mathbb{P}^n)$ be the space of all singular holomorphic foliations by curves on $\mathbb{P}^n$ ($n \geq 2$) with degree $d \geq 1.$ We show that there is subset $\mathcal{S}_d(\mathbb{P}^n)$ of…

We prove that any solution of a degenerate elliptic PDE is of class $C^1$, provided the inverse of the equation's degeneracy law satisfies an integrability criterium, viz. $\sigma^{-1} \in L^1\left (\frac{1}{\lambda} {\bf d}\lambda\right…

We compute the Hochschild and negative cyclic homology of the nodal curves, and we show that the (noncommutative) Hodge to de Rham spectral sequence degenerates at the second page. We also classify all the Hochschild classes that can be…

We consider Kempe changes on the $k$-colorings of a graph on $n$ vertices. If the graph is $(k-1)$-degenerate, then all its $k$-colorings are equivalent up to Kempe changes. However, the sequence between two $k$-colorings that arises from…

For $N \geq 4$ we classify the $(N-3)$-degenerate smooth CR maps of the three-dimensional unit sphere into the $(2N-1)$-dimensional unit sphere. Each of these maps has image being contained in a five-dimensional complex-linear space and is…

We use a counting argument and surgery theory to show that if $D$ is a sufficiently general algebraic hypersurface in $\Bbb C^n$, then any local diffeomorphism $F:X \to \Bbb C^n$ of simply connected manifolds which is a $d$-sheeted cover…

A general canonical curve X determines a finite set T(X) of hyperplanes, which is in bijective correspondence with the set of odd theta-characteristics of X. The definition of T(X) can be extended to certain singular curves, in a way that…

Let a set of nodes $\mathcal X$ in the plane be $n$-independent, i.e., each node has a fundamental polynomial of degree $n.$ Assume that\\ $\#\mathcal X=d(n,n-3)+3= (n+1)+n+\cdots+5+3.$ In this paper we prove that there are at most three…

Let X be an irreducible hypersurface in $\mathbb{P}^n$ of degree $d\geq 3$ with only isolated semi-weighted homogeneous singularities, such that $exp(\frac{2\pi i}{k})$ is a zero of the Alexander polynomial. Then we show that the…

The property of degeneration of modular graded Lie algebras, first investigated by B. Weisfeiler, is analyzed. Transitive irreducible graded Lie algebras $L=\sum_{i\in \mathbb Z}L_i,$ over an algebraically closed field of characteristic…

Consider a one-parameter family of smooth projective varieties X_t which degenerate into a simple normal crossing divisor at t=0. What is the dual variety in the limit? We answer this question for a hypersurface of degree d degenerate to…

Let k be a field. In this paper, we find necessary and sufficient conditions for a noncommutative curve of genus zero over k to be a noncommutative P^1-bundle. This result can be considered a noncommutative, one-dimensional version of…