Universal Varieties for Signatures of Membranes

Carlos Améndola; Gabriel Riffo; Leonard Schmitz

Home/Archive/arXiv

arXiv · 2610.16724

Universal Varieties for Signatures of Membranes

Carlos Améndola·Gabriel Riffo·Leonard Schmitz

View original source ↗Read source document ↗Export metadata ↓

Abstract

Membranes are sufficiently smooth vector-valued functions defined on the unit square, widely used in applications such as interpolation of image data. The signature of a membrane is a collection of nonlinear features that capture its essential geometry. We index these features in the graded Hopf algebra of two-parameter shuffle products and study the algebraic varieties for each graded component. At each truncation level, we obtain the dimension of this variety via a novel Witt-type formula. We propose a Chen-Chow theorem for membranes, provably at the second truncation level, and conjecturally for higher levels with extensive computational evidence in the computer algebra system OSCAR.

Explore related subjects

math.AG math.CO math.RA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Carlos Améndola, Gabriel Riffo, Leonard Schmitz. 2026-10-12. Universal Varieties for Signatures of Membranes. https://arxiv.org/abs/2610.16724

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Containment problems for fat point schemes via asymptotic multiplier ideals

We study a multiplier-ideal approach to the containment problem for radical ideals of points in projective space. The expected asymptotic multiplier-ideal computation depends on controlling residual base loci of the relevant symbolic linear systems. We isolate this issue in the notion of clean asymptotic multiplier ideals. Under this hypothesis, a failure of \[ I^{(2r-1)}\not\subseteq I^r \] forces \[ 2r\widehatα(I)\geq α(I^{(2r-1)})+3, \] and in particular $\widehatα(I)\geq 3$. We show that the dual-Hesse configuration is clean although its symbolic systems have a genuinely non-trivial residual asymptotic base part. We also verify cleanliness for star configurations of lines, giving a further infinite family of non-monomial examples, and formulate a cleanliness conjecture for singular loci of line arrangements.

math.AG↗

arXiv2026

An algebraic proof of Hrushovski's theorem

In his paper on the Mordell-Lang conjecture, Hrushovski employed techniques from model theory to prove the function field version of the conjecture. In doing so he was able to answer a related question of Voloch, which we refer to henceforth as Hrushovski's theorem. In this paper we shall give an alternative proof of said theorem in the characteristic $p$ setting, but using purely algebro-geometric ideas.

math.AG↗

arXiv2026

Conifold gap and all-genus mirror symmetry for local $\mathbb{P}^2$

The Conifold Gap Conjecture asserts that the polar part of the Gromov-Witten potential of a Calabi-Yau threefold near its conifold locus has a universal expression described by the logarithm of the Barnes $G$-function. In this paper, I prove the Conifold Gap Conjecture for the local projective plane. The proof relates the higher genus conifold Gromov-Witten generating series of local $\mathbb{P}^2$ to the thermodynamics of a certain statistical mechanical ensemble of repulsive particles on the positive half-line. As a corollary, this establishes the all-genus mirror principle for local $\mathbb{P}^2$ through the direct integration of the BCOV holomorphic anomaly equations.

math.AG↗