[Paper Review] On toric varieties of high arithmetical rank

Margherita Barile|ArXiv.org|Apr 12, 2005

Commutative Algebra and Its Applications7 references3 citations

TL;DR

This paper constructs a class of toric varieties in $N=2n-1$-dimensional affine space that are minimally defined by at least $N-2$ binomial equations, establishing a sharp lower bound on their arithmetical rank using étale cohomology. The key result is that for $n=2$ or $n=3$, the arithmetical rank exactly equals $N-2$, demonstrating that these varieties achieve the highest possible arithmetical rank relative to their codimension.

ABSTRACT

We describe a class of toric varieties in the $N$-dimensional affine space which are minimally defined by no less than $N-2$ binomial equations.

Motivation & Objective

• To construct a class of toric varieties in $N=2n-1$-dimensional affine space with high arithmetical rank, approaching the theoretical upper bound of $N$.
• To establish a sharp lower bound on the arithmetical rank of these varieties using étale cohomology, particularly $H^2_{ m c}(V; bZ/pbZ) \neq 0$.
• To show that the lower bound $\operatorname{ara}(V) \geq N-2$ is sharp for $n=2$ and $n=3$ by explicitly constructing set-theoretic defining equations.
• To demonstrate that the arithmetical rank is independent of the characteristic of the ground field for this class of varieties.
• To investigate whether the lower bound $\operatorname{ara}(V) \geq N-2$ remains sharp for $n \geq 4$, leaving this as an open question.

Proposed method

• Define a toric variety $V \subset K^N$ via a parametrization using $n$ variables $u_1, \dots, u_n$, with $N = 2n-1$, and monomial relations involving $d$ divisible by two distinct primes.
• Use the étale cohomology criterion from [6], Lemma 3′, to show that if $H^2_{\rm c}(V; \bbZ/p\bbZ) \neq 0$, then $\operatorname{ara}(V) \geq N-2$.
• Apply Poincaré duality to relate $H^2_{\rm c}(V)$ to $H^2_{\rm c}(K^n \setminus X)$, where $X$ is the locus $u_1 = \cdots = u_{n-1} = 0$.
• Construct an isomorphism $\tilde{\phi}: K^n \setminus X \to V \setminus \phi(X)$, showing that $H^2_{\rm c}(V \setminus \phi(X)) \simeq H^2_{\rm c}(K^n \setminus X)$.
• Use the fact that the map $\bar{\phi}: X \to \phi(X)$ induces multiplication by $d$ on $H^2_{\rm c}$, and since $p \mid d$, this map is zero on cohomology.
• Derive a non-injective map in the long exact sequence of cohomology with compact support, proving $H^2_{\rm c}(V) \neq 0$, hence $\operatorname{ara}(V) \geq N-2$.

Experimental results

Research questions

• RQ1 Can the arithmetical rank of a toric variety be bounded below using étale cohomology, particularly in high codimension?
• RQ2 Is the lower bound $\operatorname{ara}(V) \geq N-2$ sharp for toric varieties of codimension $n-1$ in $K^{2n-1}$?
• RQ3 Can the arithmetical rank be computed explicitly for $n=2$ and $n=3$ using binomial equations?
• RQ4 Does the arithmetical rank depend on the characteristic of the ground field for this class of varieties?
• RQ5 What is the arithmetical rank for $n \geq 4$, and does the bound $\operatorname{ara}(V) \geq N-2$ remain sharp?

Key findings

• The arithmetical rank of the constructed toric variety $V \subset K^N$ with $N = 2n-1$ satisfies $\operatorname{ara}(V) \geq N - 2$ for all $n \geq 2$.
• The lower bound $\operatorname{ara}(V) \geq N - 2$ is sharp for $n = 2$, where $V$ is defined by a single binomial equation: $y_1^d - x_1^{a_1 d} x_2 = 0$.
• For $n = 3$, the variety $V \subset K^5$ is set-theoretically defined by three binomials: $F_1 = y_1^d - x_1^{a_1 d} x_3$, $F_2 = y_2^d - x_2^{a_2 d} x_3$, and $G = y_1^{d-1} y_2 - x_1^{a_1(d-1)} x_2^{a_2} x_3$.
• The cohomological argument shows $H^2_{\rm c}(V; \bbZ/p\bbZ) \neq 0$ for $p \mid d$, which implies $\operatorname{ara}(V) \geq N - 2$ via the étale cohomology criterion.
• The proof is independent of the characteristic of the ground field as long as $\operatorname{char} K \neq p$, and the bound holds uniformly across characteristics.
• For $n \geq 4$, the arithmetical rank remains unknown, and the sharpness of the bound $\operatorname{ara}(V) \geq N - 2$ is left as an open problem.