Hartshorne's codimension-two complete intersection conjecture and its rank-two bundle splitting form

The famous open conjecture of Hartshorne (1974) predicts that every smooth closed subvariety of codimension two in the complex projective space $\mathbb{P}^N$ is a complete intersection for $N\ge7$, equivalently (Hartshorne's Conjecture 6.3) that every rank-two algebraic vector bundle on $\mathbb{P}^n$, $n\ge7$, splits as a direct sum of two line bundles; the community-standard strengthening asserts this already for $N\ge6$. Vatne (2009, 'Multiple Structures and Hartshorne's Conjecture') proves an equivalent scheme-theoretic reformulation: the conjecture, in the $n\ge6$ convention, holds if and only if every degree-three Cohen-Macaulay thickening $Y$ with $X\subset Y\subset X^{(1)}$, where $X=\mathbb{P}^n\subset\mathbb{P}^{n+M}$ is a linear subspace of arbitrary codimension $M$, $n\ge6$, and $X^{(1)}$ is its first infinitesimal neighborhood, contains a degree-two Cohen-Macaulay thickening $W$ with $X\subset W\subset Y$ (Conjecture 4.3 of that paper, stated in the author's open-access preprint arXiv:math/0210042). The formulation agrees with Hartshorne's primary text; the later literature contains only partial results (low-degree subvarieties of $\mathbb{P}^5$ and $\mathbb{P}^6$, toric bundles, and a fixed-degree asymptotic theorem), so the problem remains fully open.

Background

Let $k$ be an algebraically closed field, classically $k=\mathbb{C}$, and let $\mathbb{P}^N_k$ be the $N$-dimensional projective space. A closed subvariety $X\subset\mathbb{P}^N$ of codimension $c$ (that is, $\dim X=N-c$) is a complete intersection if its homogeneous ideal can be generated by exactly $c$ homogeneous polynomials; complete intersections are the subvarieties of simplest algebraic description. An algebraic vector bundle of rank two on $\mathbb{P}^N$ is, loosely, a family of two-dimensional $k$-vector spaces varying over $\mathbb{P}^N$; the building blocks are the line bundles $\mathcal{O}(d)$, and a rank-two bundle splits if it is isomorphic to a direct sum $\mathcal{O}(a)\oplus\mathcal{O}(b)$. The Serre correspondence (Proposition 6.1 and Corollary 6.2 of Hartshorne 1974) connects the two pictures: a smooth codimension-two subvariety $Y\subset\mathbb{P}^N$, $N\ge3$, whose canonical sheaf $\omega_Y$ is a multiple of the hyperplane sheaf $\mathcal{O}_Y(1)$ (such a $Y$ is called subcanonical) is the zero scheme of a section of a rank-two bundle $E$ on $\mathbb{P}^N$, determined by $Y$ up to twist, and $Y$ is a complete intersection exactly when $E$ splits; conversely, a section of any rank-two bundle vanishes along such a subvariety. Barth (1970) and Larsen (1973) proved strong cohomological connectivity of low-codimensional subvarieties, with the consequence that over $\mathbb{C}$ every smooth codimension-two $Y\subset\mathbb{P}^N$ with $N\ge6$ is automatically subcanonical; hence, over $\mathbb{C}$, the statements 'every smooth codimension-two subvariety of $\mathbb{P}^N$ is a complete intersection' and 'every rank-two vector bundle on $\mathbb{P}^N$ splits' are equivalent for every $N\ge6$. Hartshorne's general conjecture of 1974 predicts that any smooth subvariety $X\subset\mathbb{P}^n$ with $\dim X>2n/3$ is a complete intersection; the codimension-two boundary case is the celebrated open core, which Hartshorne stated as Conjecture 6.3: for $n\ge7$ there are no indecomposable (non-splitting) rank-two bundles on $\mathbb{P}^n$. Indecomposable rank-two bundles do exist on $\mathbb{P}^n$ for $n\le4$ — the Horrocks-Mumford bundle on $\mathbb{P}^4$, constructed in 1973, is the famous example — but none is known on $\mathbb{P}^n$ for any $n\ge5$, and the generally believed strengthening is that splitting holds already for $n\ge6$; Vatne's paper works in this $n\ge6$ convention. A multiple structure (also called a rope) of multiplicity $m$ on a smooth variety $X\subset\mathbb{P}^{N}$ is a scheme $Y$ with $X\subset Y$, having the same underlying variety $X$, whose multiplicity or degree — the normalized leading coefficient of the Hilbert polynomial, for a linear $X=\mathbb{P}^n$ equal to $1$ — becomes $m$ times that of $X$; $Y$ is Cohen-Macaulay if all its local rings have depth equal to dimension, the condition under which degree and duality behave well. For a linear embedding $X=\mathbb{P}^n\subset\mathbb{P}^{n+M}$ of arbitrary codimension $M$, the first infinitesimal neighborhood $X^{(1)}$, the subscheme defined by the square of the ideal of $X$, has multiplicity $M+1$ (three in the codimension-two case), and the degree-three Cohen-Macaulay structures $Y$ with $X\subset Y\subset X^{(1)}$ (that is, $I_X^2\subset I_Y\subset I_X$) form the family entering Vatne's reformulation, which ranges over all $n\ge6$ and all $M$. Vatne (2009) proves that Hartshorne's codimension-two conjecture is equivalent to the statement that every such degree-three Cohen-Macaulay structure on a linear $\mathbb{P}^n$, $n\ge6$, contains a degree-two Cohen-Macaulay thickening between $X$ and $Y$; as an illustration of the phenomena, he constructs, from the Horrocks-Mumford bundle, a degree-three Cohen-Macaulay structure on $\mathbb{P}^4\subset\mathbb{P}^{39}$, an example of the extreme embedding dimensions that such thickenings can require. The classification of multiple Cohen-Macaulay structures is itself incomplete: the Ferrand doubling construction does not produce all multiplicity-two structures on singular or certain complete-intersection supports, and the moduli of high-multiplicity, high-dimensional multiple structures are undeveloped.

Problem Statement

Prove or disprove Hartshorne's codimension-two complete intersection conjecture: over an algebraically closed field of characteristic zero (in particular over $\mathbb{C}$), every smooth closed subvariety of codimension two in $\mathbb{P}^N$ is a complete intersection for all $N\ge7$. Equivalently (Hartshorne's Conjecture 6.3): every rank-two algebraic vector bundle on $\mathbb{P}^n$, $n\ge7$, splits as a direct sum $\mathcal{O}(a)\oplus\mathcal{O}(b)$ of two line bundles. The same question in the community-standard strengthened range $N\ge6$ — where the subvariety and bundle forms remain equivalent by the Barth-Larsen theorem — and for subcanonical codimension-two subvarieties of $\mathbb{P}^5$ is part of the same open problem, but is context only and is not required for a resolution. In the equivalent form studied by Vatne (Conjecture 4.3, proved there equivalent to the codimension-two conjecture for each $n\ge6$): decide whether every degree-three Cohen-Macaulay scheme $Y$ with $X\subset Y\subset X^{(1)}$, where $X=\mathbb{P}^n\subset\mathbb{P}^{n+M}$ is a linear subspace of arbitrary codimension $M$, $n\ge7$, and $X^{(1)}$ is its first infinitesimal neighborhood, contains a degree-two Cohen-Macaulay subscheme $W$ with $X\subset W\subset Y$.

References

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