Overview
This research introduces and characterizes the polytope of effective boundary expressions associated with a divisor on the moduli space $\overline{M}_{0,n}$. The study focuses on understanding the structural properties of these polytopes and providing equivalent graph-theoretic descriptions. Computations are performed for several specific families of divisors, revealing their geometric and combinatorial characteristics.
Research Context
The work is situated within the study of divisors on $\overline{M}_{0,n}$, which represents the moduli space of stable curves of genus zero with $n$ marked points. The introduction of a polytope for effective boundary expressions provides a geometric framework for analyzing these divisors. The study also investigates the behavior of these polytopes under forgetful maps, which are operations that reduce the number of marked points on $\overline{M}_{0,n}$.
Approach
The methodology involves defining the polytope of effective boundary expressions for a divisor on $\overline{M}_{0,n}$. The research then proceeds to establish its structural properties by examining its behavior under forgetful maps that omit marked points. An equivalent description of these polytopes is developed using graph-theoretic principles. The approach includes computing these polytopes for specific families of divisors. These families include psi-classes, their pullbacks via forgetful maps, the log-canonical class, and its modifications by psi-classes. Additionally, symmetric level-one $\mathfrak{sl}_p$ conformal block divisors are considered.
Findings
- The polytope of effective boundary expressions for a divisor on $\overline{M}_{0,n}$ has definable structural properties when subjected to forgetful maps of $\overline{M}_{0,n}$ that forget marked points.
- Equivalent graph-theoretic descriptions are provided for these polytopes.
- For psi-classes and their pullbacks by forgetful maps, the corresponding polytopes are identified as unimodular simplices.
- For the log-canonical class and its modifications by psi-classes, the non-negative parts of the corresponding polytopes recover spanning forest polytopes.
- These specific polytopes also recover the subtour elimination (Held--Karp relaxation) polytope of the symmetric traveling salesman problem.
- An application of these findings is a Minkowski-like decomposition of the subtour elimination polytope into simplices.
- For symmetric level-one $\mathfrak{sl}_p$ conformal block divisors, the defining inequalities are shown to be local Turán bounds.
- The $0/1$-points of these polytopes for symmetric level-one $\mathfrak{sl}_p$ conformal block divisors correspond to balanced Turán graphs.
- Specifically, for $p=2$, these polytopes recover the perfect matching polytope.
- For $p=n/2$, these polytopes recover the fractional perfect matching polytope.
Why This Matters
The establishment of structural properties and graph-theoretic descriptions for these polytopes offers new tools for understanding divisors on $\overline{M}_{0,n}$. The recovery of known combinatorial objects, such as spanning forest polytopes and the subtour elimination polytope, within this geometric framework suggests potential for new insights into these classical problems. The Minkowski-like decomposition provides a novel structural understanding of the subtour elimination polytope. Furthermore, the connections to perfect matching and fractional perfect matching polytopes via specific conformal block divisors indicate interdisciplinary links between algebraic geometry, combinatorics, and optimization.