Two problems in arithmetic geometry inspired by homotopy theory: enriched curve counting for the Yau-Zaslow formula, and a homotopic approach to the valuative section conjecture
File(s)
Author(s)
Pajwani, Jesse
Type
Thesis
Abstract
This thesis comprises work the author has done on two separate problems in arithmetic
geometry whose origins lie in homotopy theory. The work in these problems are separate,
but linked by the underlying techniques and philosophies.
The first part of this thesis is based around an enriched version of the Yau–Zaslow
formula for counting rational curves on K3 surfaces, and is joint work with Ambrus P´al.
In order to produce this arithmetic enrichment, we first need an invariant that allows us
to count. We first construct this invariant, then relate this invariant to more classical
invariants. We then prove some basic properties about this invariant, such as its compatibility with transfer maps and Galois twists. We then focus on the question of whether
this invariant is compatible with natural power structures, and answer the question affirmitively in dimension 0. Following this, we discuss a “Fubini theorem” for this invariant
proving a version of a Fubini theorem, as well as showing that the na¨ıve version of the
Fubini theorem does not hold in this context. Finally, we pull this all together to provide
an arithmetic refinement of the Yau–Zaslow formula, closely following Beauville’s original
strategy.
The second part of this thesis is based around a variation of a result of Pop and Stix,
relating the section conjecture to valuations on our variety. We first show that we can
rephrase this result in terms of ´etale homotopy types and Berkovich spaces. We then show
that we can relate the ´etale homotopy type of a variety to its Berkovich space, before using
this to give an alternative proof of this theorem. Our alternative proof extends the result
to hold over more general fields, and we can use this flexibility to extend the theorem to
a larger class of varieties as well.
geometry whose origins lie in homotopy theory. The work in these problems are separate,
but linked by the underlying techniques and philosophies.
The first part of this thesis is based around an enriched version of the Yau–Zaslow
formula for counting rational curves on K3 surfaces, and is joint work with Ambrus P´al.
In order to produce this arithmetic enrichment, we first need an invariant that allows us
to count. We first construct this invariant, then relate this invariant to more classical
invariants. We then prove some basic properties about this invariant, such as its compatibility with transfer maps and Galois twists. We then focus on the question of whether
this invariant is compatible with natural power structures, and answer the question affirmitively in dimension 0. Following this, we discuss a “Fubini theorem” for this invariant
proving a version of a Fubini theorem, as well as showing that the na¨ıve version of the
Fubini theorem does not hold in this context. Finally, we pull this all together to provide
an arithmetic refinement of the Yau–Zaslow formula, closely following Beauville’s original
strategy.
The second part of this thesis is based around a variation of a result of Pop and Stix,
relating the section conjecture to valuations on our variety. We first show that we can
rephrase this result in terms of ´etale homotopy types and Berkovich spaces. We then show
that we can relate the ´etale homotopy type of a variety to its Berkovich space, before using
this to give an alternative proof of this theorem. Our alternative proof extends the result
to hold over more general fields, and we can use this flexibility to extend the theorem to
a larger class of varieties as well.
Version
Open Access
Date Issued
2023-06
Date Awarded
2023-10
Copyright Statement
Creative Commons Attribution NonCommercial Licence
License URL
Advisor
Pál, Ambrus
Sponsor
Engineering and Physical Sciences Research Council
Grant Number
EP/S021590/1
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)