On Fano Schemes of Complete Intersections

Abstract

We provide enumerative formulas for the degrees of varieties parameterizing hypersurfaces and complete intersections which contain projective subspaces and conics. Besides, we find all cases where the Fano scheme of the general complete intersection is irregular of dimension at least 2, and for the Fano surfaces we deduce formulas for their holomorphic Euler characteristic.

Examples of Fano manifolds with non‐pseudoeffective tangent bundle

“…For $$d=4$$, the manifold $$X$$ is isomorphic to a smooth complete intersection of two quadrics in $$\mathbb {P}^5$$. Then we have $$k=4$$ [8, Example 4.3]. For $$d=5$$, the manifold $$X$$ is isomorphic to a linear section of $$\mbox{\rm Gr(2,5)}\subset \mathbb {P}^9$$.…”