[Paper Review] On the Cox ring of blowing up the diagonal

Hendrik Bäker|arXiv (Cornell University)|Feb 22, 2014

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper computes the Cox rings of blow-ups along diagonals in products of projective spaces and in $\mathbb{P}^1$-products, providing explicit presentations via graded polynomial algebras modulo twisted Pl"ucker relations. It establishes finite generation and gives a geometric realization as an intersection of Grassmannian varieties with linear subspaces.

ABSTRACT

We compute the Cox rings of the blow-ups $\mathrm{Bl}_Δ(X' imes X')$ and $\mathrm{Bl}_Δ(\mathbb P_1^n)$ where $X'$ is a product of projective spaces and $Δ$ is the (generalised) diagonal.

Motivation & Objective

• To compute the Cox ring of the blow-up of $X' \times X'$ along its diagonal, where $X'$ is a product of projective spaces.
• To establish finite generation of the Cox ring for the blow-up of $\mathbb{P}_1^n$ along its generalized diagonal.
• To provide explicit presentations of these Cox rings in terms of generators and relations.
• To show that the total coordinate space arises as an intersection of affine Grassmannians with linear subspaces.
• To extend techniques from toric and spherical varieties to non-spherical cases, such as $\mathbb{P}_1^n$

Proposed method

• Uses the Cox construction and total coordinate space framework for normal varieties with finitely generated divisor class group.
• Applies a stretched embedding technique: embeds the original variety into a larger toric variety via addition of variables corresponding to defining equations.
• Introduces a $\mathbb{Z}^r \times \mathbb{Z}^r \times \mathbb{Z}$-grading for $X' \times X'$ and a $\mathbb{Z}^{n+1}$-grading for $\mathbb{P}_1^n$, assigning degrees to generators based on divisor class group structure.
• Employs twisted Pl"ucker relations as defining ideals: $T_{ij}T_{\infty} - T_{ik}T_{jk} + T_{il}T_{jk}$ and $T_{ij}T_{kl} - T_{ik}T_{jk} + T_{il}T_{jk}$ for $0 \leq i < j < k < l \leq n_r + 2$.
• Uses Nakayama's Lemma and primality criteria to verify that the defining ideals are prime, ensuring the Cox ring is a domain.
• Applies a lifting and restriction process via pullbacks and pushforwards of ideals under embeddings to relate the blow-up to known toric constructions.

Experimental results

Research questions

• RQ1 What is the structure of the Cox ring of the blow-up of $X' \times X'$ along its diagonal, where $X'$ is a product of projective spaces?
• RQ2 How does the Cox ring of the blow-up of $\mathbb{P}_1^n$ along its generalized diagonal behave in terms of finite generation and presentation?
• RQ3 Can the total coordinate space of such blow-ups be described as an intersection of Grassmannians with linear subspaces?
• RQ4 Under what conditions is the ideal defining the Cox ring prime, especially after blow-up and stretching?
• RQ5 How do the divisor class group and grading behave under the blow-up construction in non-spherical cases like $\mathbb{P}_1^n$?

Key findings

• The Cox ring of $\mathrm{Bl}_{\Delta_X}(X' \times X')$ is isomorphic to $R_X / I_X$, where $R_X = \mathbb{K}[T_\infty, {}_rT_{ij}]$ and $I_X$ is generated by twisted Pl"ucker relations.
• The Cox ring of $\mathrm{Bl}_{\Delta_Y}(\mathbb{P}_1^n)$ is isomorphic to $R_Y / I_Y$, with $R_Y = \mathbb{K}[S_{ij}]$ and $I_Y$ the ideal of classical Pl"ucker relations on $n+2$ variables.
• The grading of $R_X / I_X$ is explicitly given: $\deg(T_\infty) = (0,0,1)$, $\deg({}_rT_{ij}) = (e_r,0,0)$ if $j = n_r+1$, $(0,e_r,0)$ if $j = n_r+2$, and $(e_r,e_r,-1)$ otherwise.
• The grading of $R_Y / I_Y$ assigns $\deg(S_{ij}) = e_i$ if $j = n+1,n+2$ and $i \leq n$, $e_{n+1}$ if $i = n+1, j = n+2$, and $e_i + e_j - e_{n+1}$ otherwise.
• The ideal $I_X$ is prime, as shown via the primality of $I_X + \langle T_\infty \rangle$ and an application of the graded Nakayama Lemma.
• The Cox ring of $\mathrm{Bl}_{\Delta_Y}(\mathbb{P}_1^n)$ is realized as the spectrum of a quotient of a polynomial ring modulo Pl"ucker relations, with a specific $\mathbb{Z}^{n+1}$-grading that reflects the blow-up's divisor class group.