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Projective curves and weak second-order logic

Alessandro Berarducci, Francesco Gallinaro

TL;DR

The paper shows that the incidence structure of irreducible plane curves over an algebraically closed field $K$ of characteristic zero, encoded as the poset $\operatorname{Var}(K)$, is rich enough to interpret the field $K$ and, in fact, bi-interpret its two-sorted extension $(K,\mathrm{Fin}(K))$. This yields definability of arithmetic, including $\mathbb Z$, and implies undecidability of the theory in general, while for $K=\mathbb C$ a recursive axiomatization is obtained via a weak second-order framework $T_{\mathrm{Fin}}^{\mathbb C}=T_{\mathrm{rec}}\cup T_{\mathbb N}$ with stable embedding of $\mathbb N$. A central methodological innovation is a bi-interpretability between $\operatorname{Var}(K)$ and $(K,\mathrm{Fin}(K))$, built upon definable finite power sets, non-standard polynomials, and recursion on hyperfinite sets, allowing transfer of model-theoretic properties between geometry and arithmetic. The results demonstrate that the poset of algebraic curves carries a strength comparable to arithmetic theories and is sensitive to the transcendence degree of $K$, distinguishing the complex field from number fields and offering a framework for further interaction between algebraic geometry and logic.

Abstract

Given an algebraically closed field $K$ of characteristic zero, we study the incidence relation between points and irreducible projective curves, or more precisely the poset of irreducible proper subvarieties of $\mathbb P^2(K)$. Answering a question of Marcus Tressl, we prove that the poset interprets the field, and it is in fact bi-interpretable with the two-sorted structure consisting of the field $K$ and a sort for its finite subsets. In this structure one can define the integers, so the theory is undecidable. When $K$ is the field of complex numbers we can nevertheless obtain a recursive axiomatization modulo the theory of the integers. We also show that the integers are stably embedded and that the poset of irreducible varieties over the complex numbers is not elementarily equivalent to the one over the algebraic numbers.

Projective curves and weak second-order logic

TL;DR

The paper shows that the incidence structure of irreducible plane curves over an algebraically closed field of characteristic zero, encoded as the poset , is rich enough to interpret the field and, in fact, bi-interpret its two-sorted extension . This yields definability of arithmetic, including , and implies undecidability of the theory in general, while for a recursive axiomatization is obtained via a weak second-order framework with stable embedding of . A central methodological innovation is a bi-interpretability between and , built upon definable finite power sets, non-standard polynomials, and recursion on hyperfinite sets, allowing transfer of model-theoretic properties between geometry and arithmetic. The results demonstrate that the poset of algebraic curves carries a strength comparable to arithmetic theories and is sensitive to the transcendence degree of , distinguishing the complex field from number fields and offering a framework for further interaction between algebraic geometry and logic.

Abstract

Given an algebraically closed field of characteristic zero, we study the incidence relation between points and irreducible projective curves, or more precisely the poset of irreducible proper subvarieties of . Answering a question of Marcus Tressl, we prove that the poset interprets the field, and it is in fact bi-interpretable with the two-sorted structure consisting of the field and a sort for its finite subsets. In this structure one can define the integers, so the theory is undecidable. When is the field of complex numbers we can nevertheless obtain a recursive axiomatization modulo the theory of the integers. We also show that the integers are stably embedded and that the poset of irreducible varieties over the complex numbers is not elementarily equivalent to the one over the algebraic numbers.

Paper Structure

This paper contains 17 sections, 47 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Organisation of the paper
  3. Model theoretic preliminaries
  4. Shelah's theory $T^{eq}$
  5. Relative elimination of quantifiers
  6. Partial isomorphisms
  7. Weak second-order structures
  8. Definable finite power sets
  9. Recursion on sets
  10. Defining a model of $\mathsf{PA}$ inside a field of definable characteristic zero
  11. Non-standard polynomials
  12. Definable ideals
  13. The weak second-order theory of the complex field
  14. The case of transcendence degree zero
  15. A poset of algebraic curves and points
  16. ...and 2 more sections

Key Result

Proposition 2.6

Let $\varphi$ be an $L$-sentence. Suppose that for all models $\mathcal{M}, \mathcal{N}$ of $T$, $\mathcal{M} \equiv_{\Gamma} \mathcal{N}\implies \mathcal{M} \equiv_\varphi \mathcal{N}$. Then $\varphi$ is equivalent, in $T$, to a sentence in $\Gamma$.

Theorems & Definitions (140)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • proof
  • Definition 2.8
  • Definition 2.9
  • ...and 130 more