arXiv subjects
Publications and source records attributed to Lucas Braune.
We obtain a lower bound of the degree of irrationality of very general complete intersections over the complex field from the recent results of the first author and Chen--Stapleton. For combining these results, we make a minor adjustment of Chen--Stapleton's method using the trace map of differential modules.
A rational distance set is a subset of the plane such that the distance between any two points is a rational number. We show, assuming Lang's Conjecture, that the cardinalities of rational distance sets in general position are uniformly bounded, generalizing results of Solymosi-de Zeeuw, Makhul-Shaffaf, Shaffaf, and Tao. In the process, we give a criterion for certain varieties with non-canonical singularities to be of general type.
The critical loci of a map $f:X\to Y$ between smooth schemes over a field $k$ are the locally closed subschemes $Σ^i(f)\subseteq X$ where the differential of $f$ has constant rank. We prove that if $f : X\to \mathbb A^r$ is the general member of a suitably large linear family of maps from a smooth $k$-scheme $X$ to affine space, then the critical loci $Σ^i(f)$ are smooth, except in characteristic 2 where the first critical locus $Σ^1(f)$ may be singular at a finite set of points. Moreover, we compute the codimensions of the loci of second order singularities of such general maps $f :X \to \mathbb A^r$. In characteristics different from 2, the codimensions we find agree with those found by Levine in the context of differential topology. Finally, assuming that $k$ is an algebraically closed and $\dim X\ge \dim Y$, we give a local description of an arbitrary map $f :X \to Y$ at points of its first critical locus $Σ^1(f)$. In the case of functions and nondegenerate critical points, this description recovers the usual one from Morse theory.
We prove that a complete intersection of $c$ very general hypersurfaces of degree at least two in $N$-dimensional complex projective space is not ruled (and therefore not rational) provided that the sum of the degrees of the hypersurfaces is at least $\tfrac{2}{3} N + c + 1$. To this end we consider a degeneration to positive characteristic, following Kollár. Our argument does not require a resolution of the singularities of the special fiber of the degeneration. It relies on a generalization of Kollár's "algebraic Morse lemma" that controls the dimensions of the second-order Thom-Boardman singularities of general sections of Frobenius pullbacks of vector bundles.