We study the variation of linear sections of hypersurfaces in $mathbb{P}^n$. We completely classify all plane curves, necessarily singular, whose line sections do not vary maximally in moduli. In higher dimensions, we prove that the family of hyperpl
ane sections of any smooth degree $d$ hypersurface in $mathbb{P}^n$ vary maximally for $d geq n+3$. In the process, we generalize the classical Grauert-Mulich theorem about lines in projective space, both to $k$-planes in projective space and to free rational curves on arbitrary varieties.
More than four decades ago, Eisenbud, Khimv{s}iav{s}vili, and Levine introduced an analogue in the algebro-geometric setting of the notion of local degree from differential topology. Their notion of degree, which we call the EKL-degree, can be though
t of as a refinement of the usual notion of local degree in algebraic geometry that works over non-algebraically closed base fields, taking values in the Grothendieck-Witt ring. In this note, we compute the EKL-degree at the origin of certain finite covers $fcolon mathbb{A}^nto mathbb{A}^n$ induced by quotients under actions of Weyl groups. We use knowledge of the cohomology ring of partial flag varieties as a key input in our proofs, and our computations give interesting explicit examples in the field of $mathbb{A}^1$-enumerative geometry.
In a series of papers, Aluffi and Faber computed the degree of the $GL_3$ orbit closure of an arbitrary plane curve. We attempt to generalize this to the equivariant setting by studying how orbits degenerate under some natural specializations, yieldi
ng a fairly complete picture in the case of plane quartics.
