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The Chow-Lam Form

Elizabeth Pratt, Bernd Sturmfels

TL;DR

The paper develops Chow-Lam forms as a Grassmannian analogue of Chow forms, assigning to an irreducible subvariety $\mathcal{V} \subset {\rm Gr}(k,n)$ a hypersurface ${\rm CL}_\mathcal{V}$ in ${\rm Gr}(k+n-r,n)$ when $\dim \mathcal{V} = k(r-k)-1$, with degree given by the Chow-Lam degree $\lambda(\mathcal{V})$. It proves a projection formula that replaces Plücker coordinates with twistor coordinates to obtain universal equations for projected varieties, and extends the framework to matroids and positroids, yielding explicit Chow-Lam forms for several cases and a Catalan-pattern phenomenon. The paper also introduces Hurwitz-Lam forms ${\rm HL}_\mathcal{V}$ to describe branch loci of projections and defines higher Chow-Lam forms ${\rm CL}_\mathcal{V}^{(p)}$, unifying a hierarchy of coisotropic hypersurfaces with applications to amplituhedra and related combinatorial geometry. Together, these tools provide a computational and conceptual bridge between projective geometry, Schubert calculus on Grassmannians, and combinatorial structures such as matroids and positroids. The results enable universal projection computations and deepen connections to physics-inspired objects like amplituhedra.

Abstract

The classical Chow form encodes any projective variety by one equation. We here introduce the Chow-Lam form for subvarieties of a Grassmannian. By evaluating the Chow-Lam form at twistor coordinates, we obtain universal projection formulas. These were pioneered by Thomas Lam for positroid varieties in the study of amplituhedra, and we develop his approach further. Universal formulas for branch loci are obtained from Hurwitz-Lam forms. Our focus is on computations and applications in geometry.

The Chow-Lam Form

TL;DR

The paper develops Chow-Lam forms as a Grassmannian analogue of Chow forms, assigning to an irreducible subvariety a hypersurface in when , with degree given by the Chow-Lam degree . It proves a projection formula that replaces Plücker coordinates with twistor coordinates to obtain universal equations for projected varieties, and extends the framework to matroids and positroids, yielding explicit Chow-Lam forms for several cases and a Catalan-pattern phenomenon. The paper also introduces Hurwitz-Lam forms to describe branch loci of projections and defines higher Chow-Lam forms , unifying a hierarchy of coisotropic hypersurfaces with applications to amplituhedra and related combinatorial geometry. Together, these tools provide a computational and conceptual bridge between projective geometry, Schubert calculus on Grassmannians, and combinatorial structures such as matroids and positroids. The results enable universal projection computations and deepen connections to physics-inspired objects like amplituhedra.

Abstract

The classical Chow form encodes any projective variety by one equation. We here introduce the Chow-Lam form for subvarieties of a Grassmannian. By evaluating the Chow-Lam form at twistor coordinates, we obtain universal projection formulas. These were pioneered by Thomas Lam for positroid varieties in the study of amplituhedra, and we develop his approach further. Universal formulas for branch loci are obtained from Hurwitz-Lam forms. Our focus is on computations and applications in geometry.
Paper Structure (5 sections, 20 theorems, 58 equations)

This paper contains 5 sections, 20 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Coordinates and Chow Forms
  3. From Projective Space to the Grassmannian
  4. Matroids and Positroids
  5. Hurwitz-Lam Forms and Beyond

Key Result

Proposition 2.3

Suppose $dim(\mathcal{V}) = d$, and let $L$ and $M$ be linear subspaces of $\mathbb{P}^{n-1}$ such that ${\rm codim}(L \cap M) = {\rm codim}(L) + {\rm codim}(M) = d+1$. Then

Theorems & Definitions (54)

  • Example 1.1: Ruled surfaces in 3-space
  • Example 1.2: Chow-Lam form of a positroid
  • Remark 2.1
  • Example 2.2: $d=1,n=5$
  • Proposition 2.3: Intersection Formula
  • Example 2.4: Veronese surface
  • Corollary 2.5
  • proof
  • Proposition 2.6: Projection Formula
  • proof
  • ...and 44 more