Degree-one foliations on complete intersections
Mateus Figueira, Crislaine Kuster, Ruben Lizarbe, Alan Muniz
TL;DR
The paper classifies degree-one codimension-one foliations on smooth complete intersections, showing that under mild conditions the moduli space ${\sf Fol}(X,1)$ has exactly two irreducible components of logarithmic type, and that these foliations arise as restrictions from ambient projective space. A central contribution is a structure theorem for foliations on varieties covered by lines, operationalized through tangential foliations to force the dichotomy between closed rational 1-forms, algebraic integrability, or a codimension-two subfoliation of controlled degree. This framework yields the two logarithmic components $\textsf{Log}(X;1,2)$ and $\textsf{Log}(X;1,1,1)$, with the cubic threefold treated as an exception admitting a third rigid component only in related contexts. The results extend within a broader analysis of extensions/restrictions between complete intersections and ambient projective spaces, emphasizing logarithmic foliations and their restrictions. Overall, the work advances understanding of low-degree foliations on line-covered varieties and clarifies when degree-one foliations are governed by ambient logarithmic structures.
Abstract
We prove that, under mild restrictions, the space of codimension-one foliations of degree one on a smooth projective complete intersection has two irreducible components of logarithmic type. We also prove that the same conclusion holds for any smooth hypersurface of dimension at least three that is not a quadric threefold. The proof of these results follows essentially from a more general structure theorem for foliations on manifolds covered by lines.