# Surjectivity of the Wahl Map on Cubic Graphs

## Author ORCID Identifier

https://orcid.org/0000-0002-8852-9130

4-6-2023

2023

## Degree Name

Doctor of Philosophy (PhD)

## Document Type

Doctoral Dissertation

## College

Arts and Sciences

Mathematics

Dr. David Jensen

## Abstract

Much of algebraic geometry is the study of curves. One tool we use to study curves is whether they can be embedded in a K3 surface or not. If the Wahl map is surjective on a curve, that curve cannot be embedded in a K3 surface. Therefore, studying if the Wahl map is surjective for a particular curve gives us more insight into the properties of that curve. We simplify this problem by converting graph curves to dual graphs. Then the information for graphs can be used to study the underlying curves. We will discuss conditions for the Wahl map to be surjective on a cubic graph. The Wahl map on a cubic graph has two parts, one map onto the vertices and another onto the edges. We found that if the cubic graph is 3-edge-connected and non-planar, then the map on the vertices is surjective. Surjectivity of the map on the edges is not as clear. However, if the Wahl map is surjective on a cubic graph, the girth of the graph must be at least 5. Finally, based on data collected, we have observed that as the size of cubic graphs of girth at least 5 increases, the probability that the Wahl map is surjective on those graphs appears to approach 1.

## Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2023.132

COinS