Hartshorne’s complete intersection conjecture

One polynomial equation can cut out a smooth curved shape; several independent equations can cut out an intersection. In a high-dimensional complex projective space, can a smooth shape always be defined using only one equation per missing dimension when its own dimension is more than twice that number?

Write n for the shape’s dimension and c for the missing dimensions, or codimension. Hartshorne predicts that if n>2c, exactly c homogeneous equations can define it as a complete intersection. Projective coordinates identify nonzero coordinate lists that differ by an overall scale. The shape must be smooth, irreducible and nondegenerate: it has no singular points, is one algebraic piece, and does not lie in a smaller linear subspace.[1]

A smooth algebraic shape may need many equations in the form in which it is presented. Can its dimension force a presentation with the minimum number of global equations? For example, the quadric $x_0x_1+x_2x_3+x_4^2=0$ in $\mathbb P^4_{\mathbb C}$ is smooth: all five derivatives could vanish together only at the zero coordinate list, which is not a projective point. It has dimension 3 and codimension 1 and is defined by that one equation.

Projective points are nonzero coordinate lists up to common scaling. For a smooth irreducible nondegenerate $X\subset\mathbb P^{n+c}_{\mathbb C}$, let $n=\dim X$ and $c$ be its codimension. Nondegenerate means not contained in a hyperplane. With $n>2c$, must the homogeneous ideal of $X$ be generated by $c$ equations forming a regular sequence? This is the scheme-theoretic complete-intersection target. It preserves the equations’ algebraic structure, not just the same set of solutions. The quadric is an easy instance, not evidence for all higher-codimension embeddings.[1]

The claim is proved under particular degree bounds and for some codimension-two families with special geometric properties. The general problem does not impose those extra restrictions. Matching the dimension count locally is not enough: the required equations must define the entire embedded shape, with its algebraic structure.[1]

Park’s result proves the claim with additional low-degree hypotheses and treats certain codimension-two Fano/Calabi–Yau families, named for constraints on their canonical bundle. These are restrictions beyond smoothness and n>2c. Results linking complete intersections to vector-bundle splitting or to smoothability likewise do not establish the general small-codimension target.[1] [2]

A few global equations give an explicit way to construct and vary an entire family of shapes. If dimensions alone forced that description, a geometric observation would unlock algebraic tools for studying their intersections and deformations. The question concerns whether apparently complicated high-dimensional embeddings hide this simpler construction, rather than merely whether their dimensions can be counted.[1] [2]

A complete-intersection description permits the geometry to be studied through a small set of polynomial constraints and the bundles attached to them. Its reach would also limit another proposed simplification: replacing algebraic cycles by rational combinations of smooth subvarieties. Benoist and Voisin show that an asymptotic Hartshorne complete-intersection claim and universal rational smoothability cannot both be true. This page’s n>2c prediction would imply that asymptotic claim. Thus an answer here would constrain which geometric classes can be assembled from smooth pieces elsewhere, rather than only shortening a list of equations. This conditional connection does not prove or disprove either general conjecture.[1] [2]