scholarly journals Measures of irrationality for hypersurfaces of large degree

2017 ◽  
Vol 153 (11) ◽  
pp. 2368-2393 ◽  
Author(s):  
Francesco Bastianelli ◽  
Pietro De Poi ◽  
Lawrence Ein ◽  
Robert Lazarsfeld ◽  
Brooke Ullery
Keyword(s):  
Projective Space ◽  
Lower Bounds ◽  
Large Degree ◽  
Open Problems ◽  
Rational Mapping ◽  
Smooth Hypersurface ◽  
Family Of Curves

We study various measures of irrationality for hypersurfaces of large degree in projective space and other varieties. These include the least degree of a rational covering of projective space, and the minimal gonality of a covering family of curves. The theme is that positivity properties of canonical bundles lead to lower bounds on these invariants. In particular, we prove that if $X\subseteq \mathbf{P}^{n+1}$ is a very general smooth hypersurface of dimension $n$ and degree $d\geqslant 2n+1$, then any dominant rational mapping $f:X{\dashrightarrow}\mathbf{P}^{n}$ must have degree at least $d-1$. We also propose a number of open problems, and we show how our methods lead to simple new proofs of results of Ran and Beheshti–Eisenbud concerning varieties of multi-secant lines.

Mutual Embeddings

2015 ◽  
Vol 15 (01n02) ◽  
pp. 1550001
Author(s):  
ILKER NADI BOZKURT ◽  
HAI HUANG ◽  
BRUCE MAGGS ◽  
ANDRÉA RICHA ◽  
MAVERICK WOO
Keyword(s):  
Iterative Methods ◽  
Lower Bounds ◽  
Linear Equations ◽  
Graph Embedding ◽  
Open Problems ◽  
Linear Arrays ◽  
Systems Of Linear Equations

This paper introduces a type of graph embedding called a mutual embedding. A mutual embedding between two n-node graphs [Formula: see text] and [Formula: see text] is an identification of the vertices of V1 and V2, i.e., a bijection [Formula: see text], together with an embedding of G1 into G2 and an embedding of G2 into G1 where in the embedding of G1 into G2, each node u of G1 is mapped to π(u) in G2 and in the embedding of G2 into G1 each node v of G2 is mapped to [Formula: see text] in G1. The identification of vertices in G1 and G2 constrains the two embeddings so that it is not always possible for both to exhibit small congestion and dilation, even if there are traditional one-way embeddings in both directions with small congestion and dilation. Mutual embeddings arise in the context of finding preconditioners for accelerating the convergence of iterative methods for solving systems of linear equations. We present mutual embeddings between several types of graphs such as linear arrays, cycles, trees, and meshes, prove lower bounds on mutual embeddings between several classes of graphs, and present some open problems related to optimal mutual embeddings.

scholarly journals On Mixed Codes with Covering Radius $1$ and Minimum Distance $2$

10.37236/969 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Wolfgang Haas ◽  
Jörn Quistorff
Keyword(s):  
Lower Bounds ◽  
Minimum Distance ◽  
Covering Radius ◽  
Minimal Cardinality ◽  
Finite Sets ◽  
Open Problems ◽  
Latin Rectangle ◽  
Mixed Codes ◽  
Special Case

Let $R$, $S$ and $T$ be finite sets with $|R|=r$, $|S|=s$ and $|T|=t$. A code $C\subset R\times S\times T$ with covering radius $1$ and minimum distance $2$ is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality $K(r,s,t;2)$. These bounds turn out to be best possible in many instances. Focussing on the special case $t=s$ we determine $K(r,s,s;2)$ when $r$ divides $s$, when $r=s-1$, when $s$ is large, relative to $r$, when $r$ is large, relative to $s$, as well as $K(3r,2r,2r;2)$. Some open problems are posed. Finally, a table with bounds on $K(r,s,s;2)$ is given.

scholarly journals On the history of van der Waerden's theorem on arithmetic progressions

2001 ◽  
Vol 32 (4) ◽  
pp. 335-341
Author(s):  
Tom C. Brown ◽  
Jau-Shyong Peter Shiue
Keyword(s):  
Lower Bounds ◽  
Arithmetic Progressions ◽  
Upper And Lower Bounds ◽  
Open Problems ◽  
History Of

In this expository note, we discuss the celebrated theorem known as ``van der Waerden's theorem on arithmetic progressions", the history of work on upper and lower bounds for the function associated with this theorem, a number of generalizations, and some open problems.

On the dual and hessian mappings of projective hypersurfaces

1987 ◽  
Vol 101 (3) ◽  
pp. 461-468 ◽  
Author(s):  
A. D. R. Choudary ◽  
A. Dimca
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
First Order ◽  
Smooth Hypersurface ◽  
Numerical Invariants ◽  
Dual Mapping ◽  
Complex Projective

We investigate the first-order Thom–Boardman singularity sets of the dual mapping for an arbitrary (and then for a generic) smooth hypersurface in the complex projective space ℙn. Our results focus on nonemptiness, connectedness, regular stratifications and numerical invariants for these sets.

scholarly journals COMPLETE INTERSECTIONS AND MOVABLE CURVES ON THE MODULI SPACE OF SIX-POINTED RATIONAL CURVES

2012 ◽  
Vol 22 (06) ◽  
pp. 1250049
Author(s):  
PAUL L. LARSEN
Keyword(s):  
Projective Space ◽  
Complete Intersection ◽  
Moduli Space ◽  
Complex Projective Space ◽  
Projective Variety ◽  
Rational Curves ◽  
Complete Intersections ◽  
Family Of Curves ◽  
Complex Projective

A curve on a projective variety is called movable if it belongs to an algebraic family of curves covering the variety. We consider when the cone of movable curves can be characterized without existence statements of covering families by studying the complete intersection cone on a family of blow-ups of complex projective space, including the moduli space of stable six-pointed rational curves and the permutohedral or Losev–Manin moduli space of four-pointed rational curves. Our main result is that the movable and complete intersection cones coincide for the toric members of this family, but differ for the non-toric member, the moduli space of six-pointed rational curves. The proof is via an algorithm that applies in greater generality. We also give an example of a projective toric threefold for which these two cones differ.

scholarly journals Computability-theoretic complexity of effective Banach spaces

2022 ◽  
Author(s):  
◽  
Long Qian
Keyword(s):  
Banach Space ◽  
Lower Bound ◽  
Banach Spaces ◽  
Lower Bounds ◽  
Approximation Property ◽  
Upper Bounds ◽  
Space Theory ◽  
Approximation Properties ◽  
Open Problems

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

scholarly journals Spanning Trees with Many Leaves and Average Distance

10.37236/757 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ermelinda DeLaViña ◽  
Bill Waller
Keyword(s):  
Lower Bounds ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Average Distance ◽  
Independence Number ◽  
Local Independence ◽  
Open Problems ◽  
Maximum Order ◽  
Bipartite Subgraph ◽  
Number Of Leaves

In this paper we prove several new lower bounds on the maximum number of leaves of a spanning tree of a graph related to its order, independence number, local independence number, and the maximum order of a bipartite subgraph. These new lower bounds were conjectured by the program Graffiti.pc, a variant of the program Graffiti. We use two of these results to give two partial resolutions of conjecture 747 of Graffiti (circa 1992), which states that the average distance of a graph is not more than half the maximum order of an induced bipartite subgraph. If correct, this conjecture would generalize conjecture number 2 of Graffiti, which states that the average distance is not more than the independence number. Conjecture number 2 was first proved by F. Chung. In particular, we show that the average distance is less than half the maximum order of a bipartite subgraph, plus one-half; we also show that if the local independence number is at least five, then the average distance is less than half the maximum order of a bipartite subgraph. In conclusion, we give some open problems related to average distance or the maximum number of leaves of a spanning tree.

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