The Barth–Ionescu conjecture: are smooth projective varieties of degree less than their dimension complete intersections?

The Barth–Ionescu conjecture — the low-degree form of Hartshorne's famous complete-intersection conjecture — predicts that every non-degenerate smooth complex projective variety $X \subset \mathbb{P}^r$ of dimension $n$ and degree $d \leq n-1$ is a complete intersection, unless it is projectively equivalent to the Plücker-embedded Grassmannian $\mathrm{Gr}(2,5) \subset \mathbb{P}^9$ (dimension 6, degree 5). The degree bound is optimal, since the Segre embedding $\mathbb{P}^{n-1} \times \mathbb{P}^1 \subset \mathbb{P}^{2n-1}$ has degree $n$ and is not a complete intersection. Formulated by P. Ionescu around his 2008 classification of manifolds of small degree and popularized as Conjecture 3.5 of Ionescu–Russo (2013), the conjecture remains open: the best general result (Park, Math. Z. 2026) proves that such varieties are complete intersections only once $d \lesssim 0.79 \cdot n^{2/3}$, leaving a large gap to the conjectured optimal bound $d \leq n-1$.

Background

Let $\mathbb{P}^r$ be complex projective space. A positive-dimensional subvariety $X \subset \mathbb{P}^r$ is non-degenerate if it is not contained in any hyperplane; its degree $d$ is the number of intersection points with a general linear subspace of complementary dimension; its codimension is $e = r - n$, where $n = \dim X$. A complete intersection is a subvariety whose homogeneous ideal is generated by exactly $e$ homogeneous polynomials — geometrically, the intersection of $e$ hypersurfaces. Complete intersections are the 'standard' embedded varieties: their cohomology and ideal-theoretic behavior can be read off directly from the degrees of the defining equations.

Classical experience shows that smooth subvarieties of small codimension behave like complete intersections. The Barth–Larsen theorem says, roughly, that if the codimension is small compared with $n$, then $X$ is homologically similar to projective space itself; in particular its Picard group is generated by the hyperplane class. This motivated Hartshorne's celebrated conjecture (1974): a smooth subvariety $X \subset \mathbb{P}^N$ of dimension $n$ and codimension $c$ with $n \geq 2c+1$ (equivalently, codimension less than roughly $N/3$) must be a complete intersection. The conjecture is open even for $c = 2$ and even for Fano manifolds (smooth varieties whose anticanonical class is ample, the class to which the small-codimension problem reduces by adjunction-type arguments). Known positive results are of three kinds: (i) Hartshorne and Barth–Van de Ven proved that for each $n$ there is a threshold $\mathcal{N}(n)$ such that any smooth $n$-fold of degree $d \leq \mathcal{N}(n)$ (in any embedding) is a complete intersection, with $\mathcal{N}(n) \approx 0.63\sqrt{n}$; (ii) Faltings and Netsvetaev showed that varieties scheme-theoretically defined by few equations are complete intersections; (iii) Bertram–Ein–Lazarsfeld characterized complete intersections in terms of the degrees of the first defining equations, and Ran and Ballico–Chiantini settled the codimension-2 case for Fano and Calabi–Yau varieties of dimension at least 4.

Two examples calibrate the degree side. The Segre embedding $\mathbb{P}^{n-1} \times \mathbb{P}^1 \hookrightarrow \mathbb{P}^{2n-1}$ is smooth of dimension $n$ and degree $n$, and is not a complete intersection, so no threshold $\mathcal{N}(n) \geq n$ is possible. The Grassmannian $\mathrm{Gr}(2,5)$, the 6-dimensional variety parametrizing 2-dimensional subspaces of $\mathbb{C}^5$ (equivalently lines in $\mathbb{P}^4$), embeds via the Plücker embedding (sending a subspace to the wedge product of a basis of it) into $\mathbb{P}^9$ as a smooth variety of degree 5; it has Picard group $\mathbb{Z}$ and Fano index 5 (its anticanonical class is 5 times the Plücker hyperplane class), and it is not a complete intersection. Since $5 = 6 - 1$, it is the unique known obstruction at the level $d \leq n-1$.

Within this circle, Ionescu's classification program ('On manifolds of small degree', 2008, classifying linearly normal manifolds of degree at most the ambient dimension, and earlier work classifying varieties of degree at most $2e+2$) suggested the optimal form of the degree threshold, recorded as Conjecture 3.5 of Ionescu–Russo (2013) and called the Ionescu conjecture by Kwak–Park (2016) and the Barth–Ionescu conjecture by Park (2026): if $d \leq n-1$ then $X$ is a complete intersection unless it is projectively equivalent to $\mathrm{Gr}(2,5) \subset \mathbb{P}^9$. The hypothesis $d \leq n-1$ automatically places the problem inside Hartshorne's range: a non-degenerate variety has $d \geq e+1$, and an elementary argument via Ionescu's 1985 classification (recorded as a corollary in Park 2026) shows that $d \leq n-1$ forces $n \geq 2e+3$ unless $X$ is already a complete intersection of type $(2,2)$ or the Plücker $\mathrm{Gr}(2,5)$. Thus the conjecture is a genuine special ('low-degree') case of Hartshorne's conjecture — one that trades codimension information for degree information and, as Park (2026) observes, would allow one to take $\mathcal{N}(n) = n-2$.

Problem Statement

Let $X \subset \mathbb{P}^r_{\mathbb{C}}$ be a non-degenerate smooth projective variety of dimension $n \geq 1$ and degree $d$. Prove or refute the Barth–Ionescu conjecture: if $d \leq n-1,$ then $X$ is a complete intersection of $r - n$ hypersurfaces in $\mathbb{P}^r$, unless $X$ is projectively equivalent — that is, related by a linear automorphism of the ambient projective space — to the Plücker embedding $\mathrm{Gr}(2,5) \subset \mathbb{P}^9$ of the Grassmannian of 2-planes in $\mathbb{C}^5$ (the unique known exception, of dimension 6 and degree 5). In the language of Hartshorne's threshold function $\mathcal{N}(n)$ — a degree bound such that any smooth $n$-fold of degree $d \leq \mathcal{N}(n)$ (in any embedding) is a complete intersection — the conjecture would in particular allow one to take $\mathcal{N}(n) = n - 2$, with $\mathrm{Gr}(2,5)$ itself sitting at degree exactly $5 = 6 - 1$; and the bound $d \leq n-1$ is optimal, since the Segre embedding $\mathbb{P}^{n-1} \times \mathbb{P}^1 \subset \mathbb{P}^{2n-1}$ has degree $n$ and is not a complete intersection.

References

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