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Compactifications of spaces of symmetric matrices and pointed Kontsevich spaces of isotropic Grassmannians

Hanlong Fang, Alex Massarenti, Xian Wu

Abstract

We study two closely related families of varieties arising from genus $0$ stable maps to the Lagrangian Grassmannian $\operatorname{LG}(n,2n)$. First, we construct the Kausz--type compactification $\mathcal {TL}_n$ of the space of symmetric matrices and give an explicit description of its birational geometry. Second, we realize $\mathcal {TL}_n$ as a general evaluation fiber in a Kontsevich space, and then exploit this modular interpretation to derive consequences for the birational geometry of the space of pointed conics $\overline{M}_{0,1}(\operatorname{LG}(n,2n),2)$. Analogous compactifications related to orthogonal Grassmannians are also presented.

Compactifications of spaces of symmetric matrices and pointed Kontsevich spaces of isotropic Grassmannians

Abstract

We study two closely related families of varieties arising from genus stable maps to the Lagrangian Grassmannian . First, we construct the Kausz--type compactification of the space of symmetric matrices and give an explicit description of its birational geometry. Second, we realize as a general evaluation fiber in a Kontsevich space, and then exploit this modular interpretation to derive consequences for the birational geometry of the space of pointed conics . Analogous compactifications related to orthogonal Grassmannians are also presented.
Paper Structure (8 sections, 62 theorems, 170 equations, 2 figures)

This paper contains 8 sections, 62 theorems, 170 equations, 2 figures.

Table of Contents

  1. Preliminaries
  2. Kausz--Type Compactifications of Spaces of Symmetric Matrices
  3. Geometry of TL_n as Spherical Varieties
  4. Cones of divisors, positivity of anticanonical bundle, and automorphisms
  5. Cox ring and movable cone of TLn
  6. Stable Maps in Lagrangian Grassmannians
  7. Birational Geometry of Moduli Spaces of Pointed Conics in LG(n,2n)
  8. Kausz--Type Compactifications related to Orthogonal Grassmannians

Key Result

Theorem 1

Fix $n\ge 2$. Let $\mathcal{TL}_n$ be the iterated blow--up of $\operatorname{LG}(n,2n)$ constructed by starting with a transverse pair of Lagrangians $p^\pm\in \operatorname{LG}(n,2n)$ and blowing up, in a prescribed order, the strict transforms of the osculating loci at $p^+$ and $p^-$. Let $\math

Figures (2)

  • Figure 1: Schematic order of the iterated blow-ups along the osculating loci $Z_k^\pm$ producing $\mathcal{TL}_n$.
  • Figure 2: (a) A $2$--dimensional section of $\operatorname{Eff}(\mathcal{TL}_2)$. The light grey region is the section of the effective cone; the dark grey quadrilateral is the movable (and nef) cone. (b) A $2$--dimensional section of $\mathop{\mathrm{NE}}\nolimits(\mathcal{TL}_2)$ (light grey) and of the cone $\operatorname{Mov}_1(\mathcal{TL}_2)$ of moving curves (dark grey).

Theorems & Definitions (143)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Remark 1.5
  • Definition 1.6
  • ...and 133 more