My papers are also available on my arXiv page. Expository summaries of my papers will be included on my blog.
Lines on smooth cubic surfaces over Q.
In preparation, 2020. pdfabstract
We provide a complete list of all possible counts of lines on smooth cubic surfaces over the rational numbers. In particular, a smooth cubic surface over the rational numbers contains 0, 1, 2, 3, 5, 7, 9, 15, or 27 lines.
On Drużkowski's morphisms of cubic linear type.
Submitted, 2020. pdfabstractarXiv
We use theorems of Müller-Quade and Steinwandt, Scheja and Storch, and van der Waerden to study Drużkowski's morphisms of cubic linear type with invertible Jacobian. In particular, we compare the degree of such morphisms with the dimensions of various related vector spaces. These comparisons result in an inequality that, if true, would show that morphisms of cubic linear type with invertible Jacobian are injective, finite, and induce an equality of function fields.
The classical version of Bézout's Theorem gives an integer-valued count of the intersection points of hypersurfaces in projective space over an algebraically closed field. Using work of Kass and Wickelgren, we prove a version of Bézout's Theorem over any perfect field by giving a bilinear form-valued count of the intersection points of hypersurfaces in projective space. Over non-algebraically closed fields, this enriched Bézout's Theorem imposes a relation on the gradients of the hypersurfaces at their intersection points. As corollaries, we obtain arithmetic-geometric versions of Bézout's Theorem over the reals, rationals, and finite fields of odd characteristic.
All lines on a smooth cubic surface in terms of three skew lines,
with Daniel Minahan and Tianyi Zhang.
Submitted, 2020. pdfabstractarXivblog
Harris showed that the incidence variety of a smooth cubic surface containing 27 lines has solvable Galois group over the incidence variety of a smooth cubic surface containing 3 skew lines. It follows that for any smooth cubic surface, there exist formulas for all 27 lines in terms of any 3 skew lines. In response to a question of Farb, we compute these formulas explicitly. We also discuss how these formulas relate to Schläfli's count of lines on real smooth cubic surfaces.
The trace of the local A1-degree,
with Thomas Brazelton, Robert Burklund, Michael Montoro, and Morgan Opie. Homology Homotopy Appl. 23(1): 243 — 255, 2021 pdfabstractarXivblog
We prove that the local A1-degree of a polynomial function at an isolated zero with finite separable residue field is given by the trace of the local A1-degree over the residue field. This fact was originally suggested by Morel’s work on motivic transfers, and by Kass and Wickelgren’s work on the Scheja-Storch bilinear form. As a corollary, we generalize a result of Kass and Wickelgren relating the Scheja-Storch form and the local A1-degree.