Hartshorne Conjecture on Complete Intersections

Establish that any complex nondegenerate smooth projective variety X ⊂ P^{n+c} of dimension n and codimension c is a complete intersection whenever n ≥ 2c + 1.

Background

The paper studies smooth projective varieties of small codimension, with a focus on those defined by quadratic equations. A central theme is the Hartshorne conjecture, which predicts that sufficiently large dimension relative to codimension forces a variety to be a complete intersection.

While the authors (and prior work of Ionescu–Russo) verify the conjecture for quadratic varieties, they note that the conjecture remains unresolved in general. The paper includes the formal statement of the conjecture and explicitly remarks on its open status.