# P2866: Set-theoretic complete intersections for complex space curves - ID: `P2866` - Reference: `space-curves-set-theoretic-complete-intersection` - Page: https://theoremdb.org/statements/P2866 - Record maturity: Reviewed problem with recorded work ## Problem Let \(C\subset\mathbb P^3_{\mathbb C}\) be an irreducible projective curve. Must there exist homogeneous polynomials \(F,G\in\mathbb C[X_0,X_1,X_2,X_3]\) such that \(C=V(F,G)\) as sets? ### Problem setup - **Definition.** An irreducible projective curve is a one-dimensional irreducible closed subvariety of projective three-space. - **Remark.** The equality \(C=V(F,G)\) is set-theoretic: equivalently, the homogeneous ideal of \(C\) is the radical of the ideal \((F,G)\). - **Remark.** The two surfaces may be singular, reducible, and of arbitrary degrees unless the statement itself forces otherwise. ### What counts as a solution - Prove that every irreducible curve \(C\subset\mathbb P^3_{\mathbb C}\) is the set-theoretic intersection of two surfaces, with no degree restriction. - Alternatively, give an explicit irreducible complex space curve and prove that no pair of homogeneous polynomials has radical ideal equal to its homogeneous ideal. ## Status UNKNOWN as of 2026-07-31. A 2020 answer on the source page describes the characteristic-zero problem as widely open. Later checked work proves special cases and criteria, with no universal proof or complex counterexample found. Prove that every irreducible curve \(C\subset\mathbb P^3_{\mathbb C}\) is the set-theoretic intersection of two surfaces, with no degree restriction. [1](#reference-1) ## Work ### Evidence for the current status **Claim 1 (Current status and unresolved remainder).** UNKNOWN as of 2026-07-31. A 2020 answer on the source page describes the characteristic-zero problem as widely open. Later checked work proves special cases and criteria, with no universal proof or complex counterexample found. Prove that every irreducible curve \(C\subset\mathbb P^3_{\mathbb C}\) is the set-theoretic intersection of two surfaces, with no degree restriction. UNKNOWN as of 2026-07-31. A 2020 answer on the source page describes the characteristic-zero problem as widely open. Later checked work proves special cases and criteria, with no universal proof or complex counterexample found. A complete resolution must satisfy this condition: Prove that every irreducible curve \(C\subset\mathbb P^3_{\mathbb C}\) is the set-theoretic intersection of two surfaces, with no degree restriction. ### Background and intake notes Individual curve classes allow concrete ideal computations, liaison arguments, and cohomological obstructions. Exact equations and radical certificates can be stored even when they settle only one family. - Original intake status: UNKNOWN as of 2026-07-27. A 2020 answer on the source page describes the characteristic-zero problem as widely open. Later checked work proves special cases and criteria, with no universal proof or complex counterexample found. - On 2026-07-27 the MathOverflow answer and all comments were checked. They distinguish characteristic-zero uncertainty from positive-characteristic theorems and mention unresolved low-degree cases. - Cowsik and Nori's theorem in positive characteristic does not transfer to \(\mathbb C\). The base field in this record is fixed to prevent that theorem from being mistaken for a solution. - Barile and Morales, arXiv:1907.05472 and the later journal publication, study connected curves and set-theoretic complete intersections while continuing to describe the general projective-three-space question as open. - A pair of equations that contains \(C\) can have extra curve or point components. Radical equality, or a complete set-theoretic exclusion of every extra component, is required. - Trap: every curve in \(\mathbb P^3\) is an intersection of several surfaces. The target permits exactly two equations and asks for equality of their common zero set. - Recorded example: A curve that is already a scheme-theoretic complete intersection of two surfaces satisfies the condition immediately. - Recorded example: The twisted cubic has an ideal generated by three quadrics, illustrating why the set-theoretic two-equation question differs from minimal ideal generation. ### Open directions - **Route 1** (reported): Prove that every irreducible curve \(C\subset\mathbb P^3_{\mathbb C}\) is the set-theoretic intersection of two surfaces, with no degree restriction. [1](#reference-1) ### Working on this Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `space-curves-set-theoretic-complete-intersection`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome. ## References 1. Is every projective space curve a set-theoretic intersection of two surfaces?, MathOverflow question 190938. Original CC0 universal statement written after reading the answer and comments and checking later work on set-theoretic complete intersections in projective three-space. mathoverflow.net checked 2026-08-01. Original CC0 universal statement written after reading the answer and comments and checking later work on set-theoretic complete intersections in projective three-space. https://mathoverflow.net/questions/190938/is-every-projective-space-curve-a-set-theoretic-intersection-of-two-surfaces-wh - Also cited at See dataset.references[0] for the exact external source and locator. - Also cited at Editorial research route recorded 2026-07-31 - forum; reference source; checked 2026-07-31 - Source use: citation_only - Source used to formulate or check the problem record. - Source used to assess the problem's recorded status. - For Set-theoretic complete intersections for complex space curves: UNKNOWN as of 2026-07-27. A 2020 answer on the source page describes the characteristic-zero problem as widely open. Later checked work proves special cases and criteria, with no universal proof or complex counterexample found. - Source named by the research packet. 2. Robin Hartshorne and Claudia Polini, “Quasi-cyclic modules and coregular sequences”. arXiv:1907.05472 (2019). Full preprint relevant to Set-theoretic complete intersections for complex space curves. https://arxiv.org/abs/1907.05472 - preprint; reference source; arXiv:1907.05472, checked 2026-07-31; checked 2026-07-31 - Source use: citation_only - Source used to assess the problem's recorded status. - For Set-theoretic complete intersections for complex space curves: UNKNOWN as of 2026-07-27. A 2020 answer on the source page describes the characteristic-zero problem as widely open. Later checked work proves special cases and criteria, with no universal proof or complex counterexample found. 3. Robin Hartshorne and Claudia Polini, “Divisors class groups of singular surfaces”. arXiv:1301.3222 (2013). Full preprint relevant to Set-theoretic complete intersections for complex space curves. https://arxiv.org/abs/1301.3222 - preprint; reference source; arXiv:1301.3222, checked 2026-07-31; checked 2026-07-31 - Source use: citation_only - Source used to assess the problem's recorded status. - For Set-theoretic complete intersections for complex space curves: UNKNOWN as of 2026-07-27. A 2020 answer on the source page describes the characteristic-zero problem as widely open. Later checked work proves special cases and criteria, with no universal proof or complex counterexample found.