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Convex unions and completions from simplicial pseudomanifolds

Soohyun Park

TL;DR

This work develops a framework to study when unions of convex sets, realized as faces in simplicial pseudomanifolds, retain local convexity under PL homeomorphisms. It introduces a hyperplane/separation perspective on wall crossings, reduces PL moves to edge subdivisions and contractions, and characterizes the resulting contraction-space via intersections of half-spaces. A central contribution is replacing rational equivalence by linear systems of parameters to track local convexity changes, connecting these ideas to wall-relations and toric-geometry concepts. The results illuminate how edge subdivisions, external contractions, and suspensions affect convexity structures, with cross-polytopal boundaries providing canonical examples, and offer tools potentially applicable to combinatorial positivity questions in toric-like settings.

Abstract

While intersections of convex sets are convex, their unions have rather complicated behavior. Some natural contexts where they appear include duality arguments involving boundaries of convex sets and valuations, which have an Euler characteristic-like structure. However, there are certain settings where the convexity property itself is important to consider. For example, this includes (preservation of) positivity properties of divisors on toric varieties under blowdowns. In the case of (restrictions of) conormal bundles, this can be interpreted in terms of interactions between local convexity data stored in rational equivalence relations. We consider generalizations to realizations of simplicial pseudomanifolds and replace rational equivalence with effects of PL homeomorphisms. Decomposing the PL homeomorphisms into edge subdivisions and contractions, we characterize the space of suitable contraction points compatible with local convexity properties in terms of convex unions and completions. This gives rise to certain external edge subdivisions that make this ``contraction space'' of the starting edge empty, which is unexpected given the expected ``increased convexity'' from edge subdivisions. We also obtain strong affine/linear restrictions on realizations of facets containing nearby edges preserving local convexity. This implies that contracting certain nearby edges results in a very large or very small contraction space of the starting edge. As for boundary behavior, there are parallels between effects of PL homeomorphisms on induced 4-cycles in the 1-skeleton. Finally, we find effects of PL homeomorphisms and suspensions on analogues of local convexity properties stored by linear systems of parameters. This indicates that simplicial spheres PL homeomorphic to the boundary of a cross polytope store record local convexity changes in the most natural way.

Convex unions and completions from simplicial pseudomanifolds

TL;DR

This work develops a framework to study when unions of convex sets, realized as faces in simplicial pseudomanifolds, retain local convexity under PL homeomorphisms. It introduces a hyperplane/separation perspective on wall crossings, reduces PL moves to edge subdivisions and contractions, and characterizes the resulting contraction-space via intersections of half-spaces. A central contribution is replacing rational equivalence by linear systems of parameters to track local convexity changes, connecting these ideas to wall-relations and toric-geometry concepts. The results illuminate how edge subdivisions, external contractions, and suspensions affect convexity structures, with cross-polytopal boundaries providing canonical examples, and offer tools potentially applicable to combinatorial positivity questions in toric-like settings.

Abstract

While intersections of convex sets are convex, their unions have rather complicated behavior. Some natural contexts where they appear include duality arguments involving boundaries of convex sets and valuations, which have an Euler characteristic-like structure. However, there are certain settings where the convexity property itself is important to consider. For example, this includes (preservation of) positivity properties of divisors on toric varieties under blowdowns. In the case of (restrictions of) conormal bundles, this can be interpreted in terms of interactions between local convexity data stored in rational equivalence relations. We consider generalizations to realizations of simplicial pseudomanifolds and replace rational equivalence with effects of PL homeomorphisms. Decomposing the PL homeomorphisms into edge subdivisions and contractions, we characterize the space of suitable contraction points compatible with local convexity properties in terms of convex unions and completions. This gives rise to certain external edge subdivisions that make this ``contraction space'' of the starting edge empty, which is unexpected given the expected ``increased convexity'' from edge subdivisions. We also obtain strong affine/linear restrictions on realizations of facets containing nearby edges preserving local convexity. This implies that contracting certain nearby edges results in a very large or very small contraction space of the starting edge. As for boundary behavior, there are parallels between effects of PL homeomorphisms on induced 4-cycles in the 1-skeleton. Finally, we find effects of PL homeomorphisms and suspensions on analogues of local convexity properties stored by linear systems of parameters. This indicates that simplicial spheres PL homeomorphic to the boundary of a cross polytope store record local convexity changes in the most natural way.
Paper Structure (9 sections, 26 theorems, 22 equations)

This paper contains 9 sections, 26 theorems, 22 equations.

Table of Contents

  1. Background and notation
  2. (Non)separation by hyperplanes and convexity
  3. Preservation of local convexity, wall crossings, and PL homeomorphisms
  4. PL homeomorphisms and convex unions/completions
  5. Restrictions on possible realizations and convexity-preserving spaces
  6. Boundary behavior
  7. Linear systems of parameters storing analogues of local convexity changes and replacing rational equivalence
  8. Coordinate realizations and comparisons with algebraic structures
  9. PL homeomorphism-induced changes on analogues of wall relations

Key Result

Proposition 1.7

(Rational equivalence, wall relations, and convexity) (Lemma 14-1-7 on p. 421 of Mat) Let $\Sigma$ be a $(d - 1)$-dimensional simplicial fan lying inside $k^d$. Consider a wall crossing in a fan $\Sigma$ using the facets $F$ and $G$. Fix an on-wall ray $\rho \in \Sigma(1)$. Let $p \in F$ and $q \in with $c_p > 0$ with $c_q > 0$ associated to this wall crossing is a linear relation among vertices

Theorems & Definitions (68)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.5
  • Remark 1.6
  • Proposition 1.7
  • Definition 2.1
  • Proposition 2.2
  • Remark 2.3
  • proof
  • ...and 58 more