[Paper Review] Blow-ups of $\mathbb{P}^{n-3}$ at $n$ points and spinor varieties

Bernd Sturmfels, Mauricio Velasco|ArXiv.org|Jun 27, 2009

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper establishes a geometric link between the Cox ring of the blow-up of $\mathbb{P}^{n-3}$ at $n$ general points and the even spinor variety via Pfaffian generators of skew-symmetric matrices. It proves that the Cox ideal is generated by torus translates of the spinor ideal, offering a higher-dimensional generalization of Serganova-Skorobogatov's work on del Pezzo surfaces, with equality conjectured and verified computationally for $n \leq 8$. The construction uses rational functions on a moduli space to scale the spinor ideal, embedding the Cox ring into the spinor variety.

ABSTRACT

Work of Dolgachev and Castravet-Tevelev establishes a bijection between the $2^{n-1}$ weights of the half-spin representations of $\mathfrak{so}_{2n}$ and the generators of the Cox ring of the variety $X_n$ which is obtained by blowing up $\mathbb{P}^{n-3}$ at $n$ points. We derive a geometric explanation for this bijection, by embedding ${ m Cox}(X_n)$ into the even spinor variety (the homogeneous space of the even half-spin representation). The Cox ring of the blow-up $X_n$ is recovered geometrically by intersecting torus translates of the even spinor variety. These are higher-dimensional generalizations of results by Derenthal and Serganova-Skorobogatov on del Pezzo surfaces.

Motivation & Objective

• To provide a geometric explanation for the bijection between the $2^{n-1}$ generators of the Cox ring of $X_n$ and the weights of the half-spin representation of $\mathfrak{so}_{2n}$.
• To extend the representation-theoretic approach of Serganova and Skorobogatov on del Pezzo surfaces to higher-dimensional blow-ups of $\mathbb{P}^{n-3}$ at $n$ points.
• To embed the spectrum of the Cox ring of $X_n$ into the even spinor variety $S^+$ inside $\bigwedge^{\text{even}}W$.
• To show that the Cox ideal $I_X$ is generated by torus translates of the spinor ideal $I_{\text{spin}}$, with equality conjectured and verified for $n \leq 8$.

Proposed method

• The authors construct skew-symmetric $n \times n$ matrices whose subpfaffians generate the Cox ring of $X_n$, linking it to the spinor variety.
• They use Okada's identity to relate Pfaffian generators to the determinantal generators of the Cox ring, as defined in Castravet-Tevelev.
• The Cox ideal $I_X$ is shown to contain the sum of torus translates $a(c) \star I_{\text{spin}}$, where $a(c)$ are rational functions on the moduli space of point configurations.
• The construction uses a multigrading refined by the Picard group of $X_n$, with degrees indexed by $D_n$-orbits of even subsets of $[n]$.
• A quadratic Gröbner basis for $I_{\text{spin}}$ is constructed, enabling explicit computation of quadric relations in the Cox ring.
• Computational verification via Macaulay2 confirms that $I_X = I_{\text{spin}} + a(c) \star I_{\text{spin}}$ for $n \leq 8$, using a single $c \in \mathcal{G}(p)$.

Experimental results

Research questions

• RQ1 How can the bijection between the $2^{n-1}$ generators of the Cox ring of $X_n$ and the weights of the half-spin representation of $\mathfrak{so}_{2n}$ be given a geometric interpretation?
• RQ2 Can the universal torsor construction for del Pezzo surfaces be generalized to higher-dimensional blow-ups of $\mathbb{P}^{n-3}$ at $n$ points?
• RQ3 Is the Cox ideal $I_X$ of $X_n$ generated by torus translates of the spinor ideal $I_{\text{spin}}$?
• RQ4 What is the precise role of the moduli space of $n$-point configurations in parameterizing these torus actions?
• RQ5 Can the quadratic generation of $I_X$ be understood through a degeneration to Plücker monomials associated with phylogenetic trees?

Key findings

• The spectrum of the Cox ring of $X_n$ embeds into the even spinor variety $S^+$ inside $\bigwedge^{\text{even}}W$, providing a geometric realization of the Cox ring.
• The Cox ideal $I_X$ contains the sum of torus translates $\sum_{c \in \mathcal{G}(p)} a(c) \star I_{\text{spin}}$, where $a(c)$ are rational functions on the moduli space of point configurations.
• For $n \leq 8$, the equality $I_X = I_{\text{spin}} + a(c) \star I_{\text{spin}}$ holds for some $c \in \mathcal{G}(p)$, verified computationally using Macaulay2.
• The number of distinct quadratic multidegrees in $k[\bigwedge^{\text{even}}W]$ is $\lfloor n/2 \rfloor + 1$, with representatives $N_s = \deg(f_\emptyset f_{\{1,\dots,2s\}})$.
• The graded component of the Cox ring in multidegree $N_s$ has dimension $2^{s-1}$ for $s > 0$, as shown in [17, Corollary 7.4].
• The spinor ideal $I_{\text{spin}}$ is generated by quadrics, and a quadratic Gröbner basis is explicitly constructed, confirming its quadratic generation.