Non-negative polynomials on generalized elliptic curves

We study the cone of non-negative polynomials on generalized elliptic curves. We show that the zero set of every extreme ray has dense real points. If a generalized elliptic curve is embedded via a complete linear system, then we show that the convex hull of its real points (taken inside any affine chart containing all real points) is a spectrahedron. On the way, we generalize a result by Geyer–Martens on 2-torsion points in the Picard group of smooth real curves (of arbitrary genus) to possibly singular and reducible ones.

💡 Research Summary

The paper investigates the cone of non‑negative global sections of the square of a line bundle on real generalized elliptic curves. A generalized elliptic curve X over ℝ is defined as a connected projective curve with dense real points and trivial dualizing sheaf (ω_X ≅ 𝒪_X). Typical examples include reduced plane cubics and Néron n‑gons. For a line bundle L on X, the authors consider the vector space M = L⊗L and the closed convex cone
 P(X,L) = { f ∈ H⁰(X,M) : f(x) ≥ 0 for every real point x ∈ X(ℝ) }.
The first main result (Theorem 1.1) shows that any extreme ray of P(X,L) is generated by a section f whose zero set Z(f) contains a Zariski‑dense set of real points. In the smooth case this follows from the classical Riemann–Roch theorem; the authors extend the argument to singular curves by passing to the normalization π: ˜X → X, analyzing the contribution of singular points via a δ‑invariant, and using the fact that Pic⁰(˜X) is a divisible group over ℂ.
The second main result (Theorem 1.2) provides a structural description of extreme rays: there exists a coherent sheaf G (not necessarily invertible) and a positive‑semidefinite bilinear morphism ϕ : G⊗G → L⊗L such that every extreme generator f can be written as f = ϕ(g⊗g) for some g ∈ H⁰(X,G). Consequently, every element of P(X,L) is a finite sum of such squares, i.e.
 P(X,L) = { Σ_i ϕ(g_i⊗g_i) : g_i ∈ H⁰(X,G) }.
This representation yields a dual description: the dual cone P(X,L)∨ consists of linear functionals ℓ on H⁰(X,M) for which the bilinear form B_ℓ(g₁,g₂) = ℓ(ϕ(g₁⊗g₂)) is positive semidefinite.
A crucial ingredient is a divisibility result for the Picard group of real curves. Lemma 2.1 recalls that Pic⁰ of any complex curve is divisible. Proposition 2.2 shows that for a real curve X with dense real points, any divisor D supported on non‑real regular points can be written as D = 2E + div(f) where E is supported on real regular points and f is a unit that is non‑negative on all real points where it is defined. This extends the Geyer–Martens theorem on 2‑torsion in Picard groups from smooth real curves to possibly singular and reducible curves (Corollary 2.3). In particular, Lemma 2.8 proves that a generalized elliptic curve has at most two “positive” 2‑torsion classes.
Using the structural theorem, the authors address convex geometry. If X is embedded into projective space ℙⁿ by a complete linear system and H ⊂ ℙⁿ is a hyperplane disjoint from X(ℝ), then the convex hull of the real points of X, taken inside the affine chart ℝⁿ = (ℙⁿ \ H)(ℝ), is a spectrahedron (Corollary 1.3). The spectrahedral description arises from the Gram matrix of sections of G; the resulting linear matrix inequality involves block matrices of size at most ⌈(n+1)/2⌉. This is noteworthy because, while the convex hull of any curve is always a spectrahedral shadow, it is rarely a spectrahedron itself.
The proof strategy relies on the exact sequence 0 → 𝒪_X → π_*𝒪_{˜X} → S → 0, where S records the singular locus, and on the induced long exact sequence in cohomology for M. By estimating the dimension of the image of H⁰(˜X_i, M_i) under the connecting map β and bounding the singular contributions δ_i (each ≤2), the authors construct enough perturbations of a given section to guarantee the existence of the required g in Lemma 3.2. This lemma ensures that any section vanishing on a component can be altered while preserving non‑negativity, which is essential for the extreme‑ray analysis.
Overall, the paper provides a comprehensive algebraic‑geometric description of the cone of non‑negative polynomials on generalized elliptic curves, links this cone to spectrahedral convex sets, and extends classical results on Picard groups to singular real curves. The findings have implications for real algebraic geometry, semidefinite programming, and the study of real points on algebraic curves.