Tropical Chow Hypersurfaces

Abstract Given a projective variety $X\subset\mathbb{P}^n$ of codimension $k+1$, the Chow hypersurface $Z_X$ is the hypersurface of the Grassmannian $\operatorname{Gr}(k, n)$ parametrizing projective linear spaces that intersect $X$. We introduce the tropical Chow hypersurface $\operatorname{Trop}(Z_X)$. This object only depends on the tropical variety $\operatorname{Trop}(X)$ and we provide an explicit way to obtain $\operatorname{Trop}(Z_X)$ from $\operatorname{Trop}(X)$. We also give a geometric description of $\operatorname{Trop}(Z_X)$. We conjecture that, as in the classical case, $\operatorname{Trop}(X)$ can be reconstructed from $\operatorname{Trop}(Z_X)$ and prove it for the case when $X$ is a curve in $\mathbb{P}^3$. This suggests that tropical Chow hypersurfaces could be the key to construct a tropical Chow variety.

Download Full-text

Abelian varieties attached to cycles of intermediate dimension

Nagoya Mathematical Journal ◽

10.1017/s0027763000024855 ◽

1979 ◽

Vol 75 ◽

pp. 95-119 ◽

Cited By ~ 11

Author(s):

Hiroshi Saito

Keyword(s):

Abelian Variety ◽

Projective Variety ◽

Classical Case ◽

Degree Zero ◽

Algebraic Construction ◽

Jacobian Varieties ◽

Codimension One ◽

Picard Variety ◽

Complex Tori ◽

Intermediate Jacobian

The group of cycles of codimension one algebraically equivalent to zero of a nonsingular projective variety modulo rational equivalence forms an abelian variety, i.e., the Picard variety. To the group of cycles of dimension zero and of degree zero, there corresponds an abelian variety, the Albanese variety. Similarly, Weil, Lieberman and Griffiths have attached complex tori to the cycles of intermediate dimension in the classical case. The aim of this article is to give a purely algebraic construction of such “intermediate Jacobian varieties.”

Download Full-text

On the Theory of Differential Forms on Algebraic Varieties

Nagoya Mathematical Journal ◽

10.1017/s0027763000001914 ◽

1957 ◽

Vol 11 ◽

pp. 13-39

Author(s):

Yûsaku Kawahara

Keyword(s):

Function Field ◽

Projective Variety ◽

Hyperplane Section ◽

Differential Forms ◽

Classical Case ◽

Algebraic Varieties ◽

Simple Point ◽

Perfect Field ◽

Linearly Independent ◽

Normal Varieties

Let K be a function field of one variable over a perfect field k and let v be a valuation of K over k. Then is the different-divisor (Verzweigungsdivisor) of K/k(x), and (x)∞ is the denominator-divisor (Nennerdivisor) of x. In §1 we consider a generalization of this theorem in the function fields of many variables under some conditions. In §2 and §3 we consider the differential forms of the first kind on algebraic varieties, or the differential forms which are finite at every simple point of normal varieties and subadjoint hypersurfaces which are developed by Clebsch and Picard in the classical case. In §4 we give a proof of the following theorem. Let Vr be a normal projective variety defined over a field k of characteristic 0, and let ω1, …, ωs be linearly independent simple closed differential forms which are finite at every simple point of Vr. Then the induced forms on a generic hyperplane section are also linearly independent.

Download Full-text

Fibers of generic projections

Compositio Mathematica ◽

10.1112/s0010437x09004503 ◽

2010 ◽

Vol 146 (2) ◽

pp. 435-456 ◽

Cited By ~ 4

Author(s):

Roya Beheshti ◽

David Eisenbud

Keyword(s):

Positive Integer ◽

Complete Intersection ◽

Projective Variety ◽

Disjoint Union ◽

Smooth Projective Variety ◽

Higher Dimension ◽

Linear Spaces ◽

Linear Projection ◽

Dimension 2 ◽

Generic Projections

AbstractLet X be a smooth projective variety of dimension n in Pr, and let π:X→Pn+c be a general linear projection, with c>0. In this paper we bound the scheme-theoretic complexity of the fibers of π. In his famous work on stable mappings, Mather extended the classical results by showing that the number of distinct points in the fiber is bounded by B:=n/c+1, and that, when n is not too large, the degree of the fiber (taking the scheme structure into account) is also bounded by B. A result of Lazarsfeld shows that this fails dramatically for n≫0. We describe a new invariant of the scheme-theoretic fiber that agrees with the degree in many cases and is always bounded by B. We deduce, for example, that if we write a fiber as the disjoint union of schemes Y′ and Y′′ such that Y′ is the union of the locally complete intersection components of Y, then deg Y′+deg Y′′red≤B. Our method also gives a sharp bound on the subvariety of Pr swept out by the l-secant lines of X for any positive integer l, and we discuss a corresponding bound for highly secant linear spaces of higher dimension. These results extend Ran’s ‘dimension +2 secant lemma’.

Download Full-text

Strongly Uniform Bounds from Semi-Constructive Proofs

BRICS Report Series ◽

10.7146/brics.v11i31.21856 ◽

2004 ◽

Vol 11 (31) ◽

Author(s):

Philipp Gerhardy ◽

Ulrich Kohlenbach

Keyword(s):

Functional Analysis ◽

Fixed Point Theory ◽

Classical Case ◽

Point Theory ◽

Prima Facie ◽

Uniform Bounds ◽

Linear Spaces ◽

Compact Spaces ◽

Constructive Proofs ◽

First Time

In 2003, the second author obtained metatheorems for the extraction of effective (uniform) bounds from classical, prima facie non-constructive proofs in functional analysis. These metatheorems for the first time cover general classes of structures like arbitrary metric, hyperbolic, CAT(0) and normed linear spaces and guarantee the independence of the bounds from parameters raging over metrically bounded (not necessarily compact!) spaces. The use of classical logic imposes some severe restrictions on the formulas and proofs for which the extraction can be carried out. In this paper we consider similar metatheorems for semi-intuitionistic proofs, i.e. proofs in an intuitionistic setting enriched with certain non-constructive principles. Contrary to the classical case, there are practically no restrictions on the logical complexity of theorems for which bounds can be extracted. Again, our metatheorems guarantee very general uniformities, even in cases where the existence of uniform bounds is not obtainable by (ineffective) straightforward functional analytic means. Already in the purely intuitionistic case, where the existence of effective bounds is implicit, the metatheorems allow one to derive uniformities that may not be obvious at all from a given constructive proofs. Finally, we illustrate our main metatheorem by an example from metric fixed point theory.

Download Full-text

Tropical Linear Spaces and Tropical Convexity

The Electronic Journal of Combinatorics ◽

10.37236/5271 ◽

2015 ◽

Vol 22 (4) ◽

Cited By ~ 2

Author(s):

Simon Hampe

Keyword(s):

Convex Set ◽

Tropical Geometry ◽

Tropical Variety ◽

Linear Combinations ◽

Linear Spaces ◽

Classical Geometry ◽

Tropical Convexity ◽

Valuated Matroid ◽

Tropical Varieties ◽

Global Principle

In classical geometry, a linear space is a space that is closed under linear combinations. In tropical geometry, it has long been a consensus that tropical varieties defined by valuated matroids are the tropical analogue of linear spaces. It is not difficult to see that each such space is tropically convex, i.e. closed under tropical linear combinations. However, we will also show that the converse is true: Each tropical variety that is also tropically convex is supported on the complex of a valuated matroid. We also prove a tropical local-to-global principle: Any closed, connected, locally tropically convex set is tropically convex.

Download Full-text