Some examples of exceptional loci in Vojta Conjecture
Amos Turchet
TL;DR
This paper investigates the exceptional locus in Vojta's conjecture for cubic surfaces with two hyperplane sections, showing it generally consists of 21 lines but can be larger in specific cases.
Contribution
It provides an elementary analysis of the exceptional locus in a specific geometric setting, identifying when it equals 21 lines or is larger.
Findings
The exceptional locus is typically the union of 21 lines.
Examples exist where the exceptional set exceeds 21 lines.
Elementary methods suffice for the analysis.
Abstract
In this short note we discuss the exceptional locus for the Lang-Vojta's conjecture in the case of the complement of two completely reducible hyperplane sections in a cubic surface. Using elementary methods, we show that generically the exceptional set is the union of the remaining 21 lines in the surface. We also describe examples in which the exceptional set is strictly larger.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Meromorphic and Entire Functions · Analytic Number Theory Research