arXiv subjects

H. Uppal

Publications and source records attributed to H. Uppal.

5 recordsLinked to original sources

arXiv2025

Inverse curve problems on del Pezzo surfaces

We classify the number of $k$-rational lines and conic fibrations on del Pezzo surfaces over a field $k$ in terms of relatively minimal surfaces and establish rational curve analogues of the inverse Galois problem for del Pezzo surfaces. We completely solve these problems in all degrees over all global, local and finite fields and provide new solutions of the inverse Galois problem in characteristic 2. Our results generalise well-known theorems on cubic surfaces.

math.AG↗

arXiv2023

Cubic surfaces failing the integral Hasse principle

We study the integral Brauer--Manin obstruction for affine diagonal cubic surfaces, which we employ to construct the first counterexamples to the integral Hasse principle in this setting. We then count in three natural ways how such counterexamples are distributed across the family of affine diagonal cubic surfaces and how often such surfaces satisfy integral strong approximation off $\infty$.

math.NT↗

arXiv2023

Singular del Pezzo surfaces over finite fields

If $X$ is a singular del Pezzo surface of degree $d$ over a finite field $\mathbb{F}_{q}$ with only rational double point singularities, does there always exist a smooth $\mathbb{F}_{q}$-point on $X$? We show that this is true for $d\geq 3$ and give counterexamples in the case of $d=2$.

math.AG↗

arXiv2023

Integral points on affine surfaces fibered over $\mathbb{A}^{1}$

Profitant du travail de pr\'ec\'edent d'Harpaz nous utilisons la m\'ethode de descente-fibration de Swinnerton-Dyer pour \'etudier les points int\'egraux sur des surfaces affines qui sont des fibration de tores de norme 1 sur $\mathbb{A}^{1}$. Taking advantage of previous work of Harpaz we use Swinnerton-Dyer's descent-fibration method to study integral points on affine surfaces which are fibrations of norm 1 tori over $\mathbb{A}^{1}$.

math.NT↗

arXiv2022

Integral points on symmetric affine cubic surfaces

We show that if $f(u)\in \mathbb{Z}[u]$ is a monic cubic polynomial, then for all but finitely many $n\in \mathbb{Z}$ the affine cubic surface $f(u_{1})+f(u_{2})+f(u_{3})=n \subset \mathbb{A}^{3}_{\mathbb{Z}}$ has no integral Brauer-Manin obstruction to the Hasse principle.

math.NT↗