Hartshorne's conjecture: are smooth projective varieties of small codimension complete intersections?

Hartshorne's 1974 conjecture predicts that a smooth, non-degenerate complex projective variety $X \subseteq \mathbf{P}^r$ of dimension $n$ and codimension $e = r-n$ is a complete intersection whenever $n \geq 2e+1$ (equivalently, $\dim X > \tfrac{2}{3}r$), together with the borderline case $n=4$, $e=2$ (smooth 4-folds in $\mathbf{P}^6$). Its conjecturally sharp degree form, the Barth–Ionescu conjecture, asserts that degree $d \leq n-1$ forces a complete intersection except for the Plücker embedding of the Grassmannian $G(1,4) \subseteq \mathbf{P}^9$, the unique known smooth non-complete-intersection of degree $n-1$; Park's note records the uniform bound $N(n) \geq n-2$. Despite partial results — the codimension-2 Fano/Calabi–Yau case via Ran and Ballico–Chiantini, quadratic manifolds via Ionescu–Russo and Watanabe, and Park's explicit $n^{2/3}$-order degree bound improving the classical $\sqrt{n}$-order bound — both the conjecture and the optimal threshold remain open. The problem is to prove or refute these statements.

Background

Let $X \subseteq \mathbf{P}^r$ be a smooth, non-degenerate (not contained in any hyperplane) complex projective variety of dimension $n$, codimension $e = r-n$, and degree $d$ (the number of points in which a general linear subspace of dimension $e$ meets $X$). The most basic subvarieties of projective space are complete intersections: $X$ is a complete intersection when its homogeneous ideal $I_X \subseteq \mathbb{C}[x_0,\dots,x_r]$, the ideal generated by all homogeneous polynomials vanishing on $X$, can be generated by exactly $e$ homogeneous polynomials, so that $X$ is cut out scheme-theoretically — by its full system of defining equations, not merely as a point set — by $e$ hypersurfaces. Complete intersections are the subvarieties whose invariants (cohomology, canonical class, Hilbert polynomial) are directly computable, so the question of how far a smooth subvariety of projective space can be from one is a central theme of projective algebraic geometry.

In the early 1970s, Barth and Hartshorne independently discovered that a smooth variety of small codimension is homologically like a complete intersection: Barth's theorems identify the cohomology of $X$ with that of $\mathbf{P}^r$ in low degrees. Since the only smooth projective manifolds of small codimension anyone knew were complete intersections, Hartshorne was led to his famous conjecture: if $n \geq 2e+1$, equivalently $\dim X > \tfrac{2}{3}\,r$, then $X$ is a complete intersection. The conjecture is customarily stated together with its borderline case $n=4$, $e=2$, that is, smooth 4-folds in $\mathbf{P}^6$ (the case $\dim X = \tfrac{2}{3}\,r$); this is the form recorded in Park's note. Fifty years later the conjecture is open already in codimension $2$.

The main partial results are as follows. Ran proved that a smooth codimension-2 subvariety of $\mathbf{P}^r$, $r \geq 6$, that is subcanonical (its canonical class, the divisor class of top-degree holomorphic differential forms, is an integral multiple of the hyperplane class) is a complete intersection; on this basis Ballico–Chiantini verified the conjecture for codimension $2$ when $X$ is a Fano variety (anticanonical class $-K_X$ ample) or a Calabi–Yau variety ($K_X$ trivial), while for codimension $e \geq 3$ the conjecture remains open even for Fano varieties. Park gave a new, shorter proof of the codimension-2 Fano/Calabi–Yau case using generic projections (projection from a general linear center disjoint from $X$) and the Evans–Griffith syzygy theorem (a splitting criterion for vector bundles on projective space). Ionescu–Russo proved the conjecture for quadratic manifolds (smooth varieties whose homogeneous ideal is generated by quadrics) with $n \geq 2e+1$ and classified the boundary layer $n = 2e$; Watanabe recently classified the layer $n = 2e-1$.

There is also a degree-theoretic formulation. Hartshorne proved that some function $N(n)$ exists such that $d \leq N(n)$ forces $X$ to be a complete intersection, and an explicit bound of order $\sqrt{n}$ was obtained in the 1970s. Park improved the explicit bound to

$$N(n) \;\geq\; \min\Bigl\{\, n-2, \ \sqrt[3]{(n+2)^2/2}+2 \,\Bigr\},$$

of order $n^{2/3}$. Two classical examples show roughly where the truth must lie: the Segre embedding $\mathbf{P}^{n-1}\times\mathbf{P}^1 \subseteq \mathbf{P}^{2n-1}$ (for $n \geq 3$) has dimension $n$ and degree $n$ but is not a complete intersection, and the Plücker embedding $G(1,4) \subseteq \mathbf{P}^9$ of the Grassmannian parametrizing lines in $\mathbf{P}^4$ has dimension $6$, codimension $3$, degree $5$, and is not a complete intersection. These motivate the Barth–Ionescu conjecture (recorded as Conjecture 3.5 of Ionescu–Russo): $d \leq n-1$ should force a complete intersection except for $G(1,4) \subseteq \mathbf{P}^9$, the unique known smooth non-complete-intersection of degree $n-1$. In terms of the optimal forcing function this predicts $N(n) = n-1$ for every $n \geq 3$ with $n \neq 6$, the single value $N(6) = 4 = n-2$ being forced down by the Grassmannian itself; Park records the uniform consequence $N(n) \geq n-2$. Park also observed (Corollary 2.5 of his note) that $d \leq n-1$ implies $n \geq 2e+1$ unless $X$ is already a complete intersection or is the Grassmannian example, so the Barth–Ionescu conjecture is a special case of Hartshorne's conjecture: the two statements are two windows on one question — when does smoothness force a complete intersection?

Problem Statement

Resolve Hartshorne's conjecture on complete intersections, in the form recorded in Park's note, together with its conjecturally sharp degree form.

Let $X \subseteq \mathbf{P}^r$ be a smooth, non-degenerate (not contained in any hyperplane) projective variety over $\mathbb{C}$ of dimension $n$, codimension $e = r-n$, and degree $d$.

(1) Hartshorne's conjecture. Prove, or refute by an explicit example, that if $n \geq 2e+1$, or if $n = 4$ and $e = 2$, then $X$ is a complete intersection, i.e. the homogeneous ideal $I_X \subseteq \mathbb{C}[x_0,\dots,x_r]$ of all polynomial equations vanishing on $X$ is generated by exactly $e$ elements (a scheme-theoretic complete intersection).

(2) The sharp degree form (Barth–Ionescu conjecture). Prove, or refute by an explicit example, that $d \leq n-1$ implies that $X$ is a complete intersection unless $X$ is projectively equivalent to the Plücker embedding $G(1,4) \subseteq \mathbf{P}^9$ of the Grassmannian parametrizing lines in $\mathbf{P}^4$, the unique known smooth non-complete-intersection of degree $n-1$. Writing $N(n)$ for the largest integer such that every smooth non-degenerate $n$-dimensional variety $X \subseteq \mathbf{P}^r$ (for any $r$) of degree at most $N(n)$ is a complete intersection — Hartshorne proved that such a finite function exists — a positive answer would pin the optimal values exactly: $N(n) = n-1$ for every $n \geq 3$ with $n \neq 6$, and $N(6) = 4 = n-2$, the single dimension lowered by the Grassmannian itself; Park records the uniform formulation $N(n) \geq n-2$, and proves $N(n) \geq \min\{n-2, \sqrt[3]{(n+2)^2/2}+2\}$.

The two claims share one scientific objective: deciding exactly when smoothness together with small codimension — equivalently, by Park's observation, small degree — forces a complete intersection. Claim (2) is a special case of claim (1): by Corollary 2.5 of Park's note, $d \leq n-1$ implies $n \geq 2e+1$ unless $X$ is already a complete intersection or is the Grassmannian example, so a proof of (1) settles (2) and pins the optimal threshold values above, with the Plücker Grassmannian as the unique exception at degree $n-1$ and the Segre embeddings $\mathbf{P}^{n-1}\times\mathbf{P}^1 \subseteq \mathbf{P}^{2n-1}$ of degree $n$ showing sharpness at degree $n$. A single explicit counterexample satisfying the hypotheses of claim (1) refutes Hartshorne's conjecture; a counterexample to claim (2), which by the same corollary automatically satisfies $n \geq 2e+1$, refutes both claims. Partial resolutions restricted to special classes of varieties (for example quadratic manifolds, or Fano and Calabi–Yau varieties in codimension 2) leave the general conjecture and the optimal threshold open.

References

1. Jinhyung Park, Some remarks on smooth projective varieties of small degree and codimension, Mathematische Zeitschrift 312, no. 3 (2026); preprint arXiv:2405.11933 (2024), DOI: 10.1007/s00209-026-03965-1, https://arxiv.org/abs/2405.11933
2. Robin Hartshorne, Varieties of small codimension in projective space, Bulletin of the American Mathematical Society 80 (1974), 1017–1032, DOI: 10.1090/s0002-9904-1974-13612-8, https://doi.org/10.1090/s0002-9904-1974-13612-8
3. Ziv Ran, On projective varieties of codimension 2, Inventiones Mathematicae 73 (1983), 333–336, DOI: 10.1007/BF01394029, https://doi.org/10.1007/BF01394029
4. Edoardo Ballico and Luca Chiantini, On smooth subcanonical varieties of codimension 2 in P^n, n ≥ 4, Annali di Matematica Pura ed Applicata (4) 135 (1983), 99–117, DOI: 10.1007/BF01781064, https://doi.org/10.1007/BF01781064
5. Paltin Ionescu and Francesco Russo, Manifolds covered by lines and the Hartshorne conjecture for quadratic manifolds, American Journal of Mathematics 135, no. 2 (2013), 349–360, DOI: 10.1353/ajm.2013.0020, https://doi.org/10.1353/ajm.2013.0020 (arXiv:1209.2047)
6. Kiwamu Watanabe, Quadratic varieties of small codimension, SIGMA — Symmetry, Integrability and Geometry: Methods and Applications 21 (2025), 045, DOI: 10.3842/SIGMA.2025.045, https://doi.org/10.3842/SIGMA.2025.045 (arXiv:2405.04002)
7. Supravat Sarkar, Smooth blow up structures on projective bundles, arXiv:2412.00021 (2024); published as Smooth blow up structures on projective bundles over projective spaces in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (2025), DOI: 10.2422/2036-2145.202501_023, https://arxiv.org/abs/2412.00021