\input{"preamble.tex"}

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\title{
\textbf{
    An introduction to \(\AA^1\) homotopy theory using enumerative
    examples
  }
    \\ {\normalsize Kirsten Wickelgren, IAS/PCMI GSS 2021} \\
  }







\begin{document}

\date{}
\maketitle
\begin{flushleft}
\textit{D. Zack Garza} \\
\textit{University of Georgia} \\
  \textit{\href{mailto: dzackgarza@gmail.com}{dzackgarza@gmail.com}} \\
{\tiny \textit{Last updated:} 2021-07-27 }
\end{flushleft}


\newpage

% Note: addsec only in KomaScript
\addsec{Table of Contents}
\tableofcontents
\newpage

\hypertarget{kirsten-wickelgren-talk-1-wednesday-july-14}{%
\section{Kirsten Wickelgren, Talk 1 (Wednesday, July
14)}\label{kirsten-wickelgren-talk-1-wednesday-july-14}}

\hypertarget{intro}{%
\subsection{Intro}\label{intro}}

\textbf{Abstract}:

\begin{quote}
Morel and Voevodsky's \({\mathbb{A}}^1\) homotopy theory imports tools
from algebraic topology into the study of schemes, or in other words,
into the study of the solutions to polynomial equations. This theory
produces greater understanding of arithmetic and geometric aspects these
solutions. We will introduce some of this theory using as a guide
questions such as ``How many lines meet 4 lines in 3-space?''
\end{quote}

\textbf{References/Background}:

\begin{itemize}
\tightlist
\item
  Some algebraic topology or algebraic geometry, for example as
  described in Hatcher's and Hartshorne's books
\item
  \href{https://www.ias.edu/sites/default/files/L1\%20-\%20Intro\%20to\%20A1-homotopy\%20theory\%20using\%20enumerative\%20examples\%20-\%20Jul\%2013\%2C\%202021_1.pdf}{Lecture
  Notes 1}
\item
  \href{https://www.ias.edu/sites/default/files/L2\%20-\%20Intro\%20to\%20A1-homotopy\%20using\%20enumerative\%20examples\%20-\%20Jul\%2015\%2C\%202021.pdf}{Lecture
  Notes 2}
\item
  \href{https://www.ias.edu/sites/default/files/Pauli-Exercises_PCMI.pdf}{Exercises}
\end{itemize}

\begin{remark}

Enumerative geometry counts algebro-geometric objects over
\({\mathbb{C}}\). Example: how many lines meet 4 generic lines in
\({\mathbb{P}}^3\)? The answer is 2, and our goal is to record this kind
of arithmetic information about geometric objects over a field \(k\)
whose intersections are fixed over
\(\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\) but not
necessarily \(k\) itself. Our main tool will be
\({\mathbb{A}}^1{\hbox{-}}\)homotopy theory, due to Morel-Voevodsky.

\end{remark}

\hypertarget{classical-theory}{%
\subsection{Classical Theory}\label{classical-theory}}

\begin{remark}

First some classical homotopy theory. The sphere can be defined as
\begin{align*}
S^n \coloneqq\left\{{{ {x}_1, {x}_2, \cdots, {x}_{n}} {~\mathrel{\Big|}~}\sum x_i^2 = 1}\right\} \simeq{\mathbb{P}}^n({\mathbb{R}}) / {\mathbb{P}}^{n-1}({\mathbb{R}})
,\end{align*}
and we have a degree map \([S^n, S^n] \to {\mathbb{Z}}\). Given any
\(f\in {\mathsf{Top}}(S^n, S^n)\) and \(p\in S^n\), we can write
\(f^{-1}(p) = \left\{{{ {q}_1, {q}_2, \cdots, {q}_{N}}}\right\}\) and
compute \(\deg f = \sum_{i=1}^N \deg_{q_i} f\) in terms of \textbf{local
degrees}. Letting \(V\) be a ball containing \(p\), we have
\(F^{-1}(V) \supseteq U \ni q_i\) another ball such that
\(U \cap f^{-1}(p) = q_i\). Then
\(U/{{\partial}}U \simeq S^n \simeq V/{{\partial}}V\), so we can define
a map
\begin{align*}
\mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu: {U \over U\setminus\left\{{q_i}\right\}} \to {V \over V\setminus\left\{{p}\right\}}
\end{align*}
and define
\(\deg_{q_i} f \coloneqq\deg \mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu\).

Letting \({\left[ {{ {x}_1, {x}_2, \cdots, {x}_{n}}} \right]}\) be
oriented coordinates about \(q_i\) and
\({\left[ {{ {y}_1, {y}_2, \cdots, {y}_{n}}} \right]}\) about \(p\),
then
\(f = {\left[ {{ {f}_1, {f}_2, \cdots, {f}_{n}}} \right]}: {\mathbb{R}}^n\to {\mathbb{R}}^n\)
and we can consider
\(J_f \coloneqq\operatorname{det}\qty{{\partial}f_i \over {\partial}x_i}\).
There is then a formula
\begin{align*}
\deg_{q_i}(f) = 
\begin{cases}
+1 & J_f(q_i) > 0 
\\
-1 & J_f(q_i) < 0.
\end{cases}
,\end{align*}
and for all \(q_i\) we have \(\deg f = \# f^{-1}({\operatorname{pt}})\),
i.e.~the number of solutions to the polynomial system
\(\left\{{f_1 = f_2 = \cdots f_n = 0}\right\}\).

\end{remark}

\begin{example}[?]

If \(f\in {\mathbb{C}}[x]\) of degree \(n\), we can regard \(f\) as a
function
\(f: {\mathbb{P}}^1({\mathbb{C}})\to {\mathbb{P}}^1({\mathbb{C}})\) and
by the fundamental theorem of algebra,
\begin{align*}
\deg f = n = \# \left\{{f=0}\right\}
.\end{align*}

\end{example}

\begin{remark}

We can similarly count solutions to \(f=0\) when \(f\) is a section of a
rank \(n\) vector bundle

\begin{center}
\begin{tikzcd}
V
\ar[d, "p"] 
\\
X 
\ar[u, bend left, dotted, "f"]
\end{tikzcd}
\end{center}

This count can be computed using the Euler class:
\begin{align*}
e(V) = e(V, f) = \sum_{q_i\in \left\{{f=0}\right\}} \deg q_i f
.\end{align*}

\end{remark}

\begin{example}[?]

Let \(X \coloneqq{\operatorname{Gr}}(1, 3)_{/{\mathbb{C}}}\), the
Grassmannian parameterizing dimension 2 subspaces
\(W \subseteq {\mathbb{C}}^4\), or equivalently lines in
\({\mathbb{P}}W \subseteq {\mathbb{P}}({\mathbb{C}}^4) \cong {\mathbb{P}}^3({\mathbb{C}})\),
where \({\mathbb{P}}W\) is defined as \(W\setminus\left\{{0}\right\}\)
where \(\lambda w\sim w\) for any \(\lambda \in {\mathbb{C}}^{\times}\).
The tautological is a rank 2 bundle:

\begin{center}
\begin{tikzcd}
S_{[{\mathbb{P}}W]} = W
  \ar[r] 
& 
S
  \ar[d] 
\\
& 
{\operatorname{Gr}}(1, 3)_{/ {{\mathbb{C}}}} 
\end{tikzcd}
\end{center}

Let \(L_1, \cdots, L_4\) be four lines in \({\mathbb{P}}^3\), then
\(\left\{{\text{lines intersecting all } L_i}\right\} = \left\{{f=0}\right\}\)
where \(f\) is a section (depending on the \(L_i\)) of the bundle

\begin{center}
\begin{tikzcd}
{\mathcal{E}}\coloneqq\qty{\bigwedge\nolimits^2 S^*}^{\oplus 4}\
\ar[d] 
\\
{\operatorname{Gr}}(1, 3)
\end{tikzcd}
\end{center}

and the Euler number of this bundle counts the number of such
intersections. In particular, \(e({\mathcal{E}})\) is independent of the
choice of lines and section, provided they're sufficiently generic (so
the \(L_i\) do not pairwise intersect). Using the splitting principle
and knowledge of \(H^*({\operatorname{Gr}})\), one can compute
\(e({\mathcal{E}}) = 2\).

\end{example}

\hypertarget{over-arbitrary-fields-grothendieck-witt}{%
\subsection{Over arbitrary fields:
Grothendieck-Witt}\label{over-arbitrary-fields-grothendieck-witt}}

\begin{remark}

We'd like to do this over arbitrary fields \(k\). Lannes and Morel
defined degrees for rational maps
\(f:{\mathbb{P}}^1 \to {\mathbb{P}}^1\). Above we only remembered the
sign of \(J_f\), and here we'll allow remembering more: \(\deg f\) will
be valued in \({\operatorname{GW}}(k)\). We can realize
\({\operatorname{GW}}(k)\) as the group completion of the semiring of
nondegenerate symmetric bilinear forms under \(\perp, \otimes_k\), where
we complete with respect to \(\perp\). It is related to the Witt group
by
\begin{align*}
W(K) { { \, \xrightarrow{\sim}\, }}{{\operatorname{GW}}(k) \over {\mathbb{Z}}\left\langle{q_{ {\operatorname{hyp}}} }\right\rangle } \coloneqq{{\operatorname{GW}}(k) \over {\mathbb{Z}} { \left[ {\left\langle{1}\right\rangle + \left\langle{-1}\right\rangle} \right] } }
.\end{align*}
There is a \textbf{rank map}
\begin{align*}
\operatorname{rank}: {\operatorname{GW}}(k) &\to {\mathbb{Z}}\\
q: (V{ {}^{ \scriptstyle\otimes_{k}^{2} }  } \to K) &\mapsto \dim_k V
,\end{align*}
which can be realized by a pullback

\begin{center}
\begin{tikzcd}
    {{\operatorname{GW}}(k)} && {W(k)} & {} \\
    \\
    {\mathbb{Z}}&& {{\mathbb{Z}}/2}
    \arrow["\operatorname{rank}", from=1-3, to=3-3]
    \arrow["\operatorname{rank}", from=1-1, to=3-1]
    \arrow[from=3-1, to=3-3]
    \arrow[from=1-1, to=1-3]
    \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMCwwLCJcXEdXKGspIl0sWzMsMF0sWzIsMCwiVyhrKSJdLFswLDIsIlxcWloiXSxbMiwyLCJcXFpaLzIiXSxbMiw0LCJcXHJhbmsiXSxbMCwzLCJcXHJhbmsiXSxbMyw0XSxbMCwyXSxbMCw0LCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0=}{Link
to Diagram}
\end{quote}

We can also write \({\operatorname{GW}}(k)\) in terms of generators
\(\left\langle{a}\right\rangle\) where
\(a\in k^{\times}/ (k^{\times})^{\times 2}\), where
\(\left\langle{a}\right\rangle\) is associated to a bilinear form
\begin{align*}
\left\langle{a}\right\rangle: k^{\times 2} &\to k \\
(x, y) &\mapsto axy
,\end{align*}
subject to relations

\begin{itemize}
\tightlist
\item
  \(\left\langle{a}\right\rangle\left\langle{b}\right\rangle = \left\langle{ab}\right\rangle\)
\item
  \(\left\langle{u}\right\rangle + \left\langle{v}\right\rangle = \left\langle{uv(u+v)}\right\rangle + \left\langle{u+v}\right\rangle\)
\item
  \(\left\langle{u}\right\rangle + \left\langle{-u}\right\rangle = \left\langle{1}\right\rangle + \left\langle{-1}\right\rangle = q_{\operatorname{hyp}}\),
  which is the matrix
  \begin{align*}
  q_{\operatorname{hyp}}= {
  \begin{bmatrix}
    {0} & {1} 
  \\
    {1} & {0}
  \end{bmatrix}
  }
  .\end{align*}
\end{itemize}

\end{remark}

\begin{example}[of known $\GW$ groups]

The \textbf{signature} is the difference between the numbers of positive
and negative ones in the associated matrix, and one can show
\begin{align*}
\operatorname{rank}: {\operatorname{GW}}({\mathbb{C}}) &{ { \, \xrightarrow{\sim}\, }}{\mathbb{Z}}\\ 
(\operatorname{rank}, \operatorname{sig}): {\operatorname{GW}}({\mathbb{R}}) &{ { \, \xrightarrow{\sim}\, }}{\mathbb{Z}}^{\times 2} \\
(\operatorname{rank}, {\operatorname{disc}}): {\operatorname{GW}}({\mathbb{F}}_q) &{ { \, \xrightarrow{\sim}\, }}{\mathbb{Z}}\times{\mathbb{F}}_q^{\times}/ ({\mathbb{F}}_q^{\times})^{\times 2}
\end{align*}
where the last is a situation where we can compute étale cohomology.

\end{example}

\begin{example}[Springer's theorem]

Let \(k\in \mathsf{Field}\) be complete and discretely valued with
residue field \(\kappa\), e.g.~\(k= { {\mathbb{Q}}_p }\) or
\({\mathbb{F}}_p{\left(\left( t \right)\right) }\) with
\(\kappa = {\mathbb{F}}_p\). Then if \(\operatorname{ch}k \neq 2\),
\begin{align*}
GW(k) { { \, \xrightarrow{\sim}\, }}W(k){ {}^{ \scriptscriptstyle\oplus^{2} }  }
.\end{align*}

\end{example}

\begin{remark}

For \(E_{/ {k}}\) a finite separable field extension, we'll have
transfers
\begin{align*}
\operatorname{Tr}_{E_{/ {k}} }: {\operatorname{GW}}(E) &\to {\operatorname{GW}}(k) \\
(V{ {}^{ \scriptstyle\otimes_{k}^{2} }  } \xrightarrow{\beta} E) &\mapsto (V{ {}^{ \scriptstyle\otimes_{k}^{2} }  } \xrightarrow{\beta} E \xrightarrow{\operatorname{Tr}_{E_{/k}}} k)
,\end{align*}
which coincide with classical transfers for field extensions.

\end{remark}

\begin{remark}

For Lannes/Morel's formula, given
\({\mathbb{P}}^1_{/ {k}} \xrightarrow{f} {\mathbb{P}}^1_{/ {k}}\) and
\(p\in {\mathbb{P}}^1_{/ {k}}\), we can write
\(f^{-1}(p) = \left\{{{ {q}_1, {q}_2, \cdots, {q}_{N}}}\right\}\) and
suppose \(J(q_i) = f'(q_i) \neq 0\) for all \(i\). Then we remember the
entire Jacobian and set
\begin{align*}
\deg f \coloneqq\sum_{i=1}^N \operatorname{Tr}_{k(q_i)_{/ {k}} } \left\langle{J(q_i)}\right\rangle
,\end{align*}
which in fact doesn't depend on \(p\). Morel defines an
\({\mathbb{A}}^1{\hbox{-}}\)degree
\begin{align*}
\deg^{{\mathbb{A}}^1}: [{\mathbb{P}}^n/{\mathbb{P}}^{n+1}, {\mathbb{P}}^n/{\mathbb{P}}^{n+1}]^{{\mathbb{A}}^1} \to {\operatorname{GW}}(k)
,\end{align*}
where we are taking unstable \({\mathbb{A}}^1{\hbox{-}}\)homotopy
classes of maps. Noting that an element of
\({\operatorname{GW}}({\mathbb{R}})\) was determined by its rank and
signature, we get a commutative diagram showing that
\(\deg^{{\mathbb{A}}^1}\) is compatible with rank, signature, and the
classical algebraic topological degree. There are other ways of
computing this degree besides taking the above sum: Cazanave,
\href{https://arxiv.org/abs/2103.16614}{Brazelton-McKean-Pauli} give
formulas in terms of Bézoutians.

\end{remark}

\hypertarget{homotopy}{%
\subsection{Homotopy}\label{homotopy}}

\begin{remark}

Recall that
\begin{align*}
X\wedge Y \coloneqq{ X\times Y \over (X\times{\operatorname{pt}}) \cup({\operatorname{pt}}\times Y)} && \in {\mathsf{Top}}_*
,\end{align*}
and \(S^n \wedge S^m \xrightarrow{\sim} S^{n+m}\) and
\((S^1){ {}^{ \scriptscriptstyle\wedge^{n} } } \xrightarrow{\sim} S^n\),
so we define
\begin{align*}
{\Sigma}_{S^1} X \coloneqq S^1 \wedge X
.\end{align*}
In \({\mathbb{A}}^1\) homotopy theory we declare
\({\mathbb{A}}^1 \simeq{\operatorname{pt}}\).

\end{remark}

\begin{example}[?]

We can take a pushout of the following form:

\begin{center}
\begin{tikzcd}
    & z && z \\
    z & {{\mathbb{G}}_m} && {{\mathbb{A}}^1\simeq{\operatorname{pt}}} \\
    \\
    {1/z} & {{\operatorname{pt}}\simeq{\mathbb{A}}^1} && { \therefore {\mathbb{P}}^1 \simeq{\Sigma}_{S^1} {\mathbb{G}}_m}
    \arrow[maps to, from=2-1, to=4-1]
    \arrow[from=2-2, to=4-2]
    \arrow[from=2-2, to=2-4]
    \arrow[maps to, from=1-2, to=1-4]
    \arrow[from=4-2, to=4-4]
    \arrow[from=2-4, to=4-4]
    \arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=4-4, to=2-2]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsOCxbMSwxLCJcXEdHX20iXSxbMSwzLCJcXHB0IFxcaG9tb3RvcGljIFxcQUFeMSJdLFswLDEsInoiXSxbMCwzLCIxL3oiXSxbMywxLCJcXEFBXjFcXGhvbW90b3BpYyBcXHB0Il0sWzEsMCwieiJdLFszLDAsInoiXSxbMywzLCJcXFBQXjEgXFxob21vdG9waWMgXFxTdXNwZW5kX3tTXjF9IFxcR0dfbSJdLFsyLDMsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifX19XSxbMCwxXSxbMCw0XSxbNSw2LCIiLDIseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJtYXBzIHRvIn19fV0sWzEsN10sWzQsN10sWzcsMCwiIiwxLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV1d}{Link
to Diagram}
\end{quote}

Here the formalism of homotopy pushouts allows us to conclude that in an
appropriate \({\mathbb{A}}^1{\hbox{-}}\)homotopy category,
\begin{align*}
{\Sigma}_{S^1} {\mathbb{G}}_m \coloneqq S^1\wedge{\mathbb{G}}_m \simeq{\mathbb{P}}^1
.\end{align*}

\end{example}

\begin{remark}[on motivic spheres]

We have
\begin{align*}
{\mathbb{G}}_m \coloneqq\operatorname{Spec}k[z, 1/z] = {\mathbb{A}}^1\setminus\left\{{{\operatorname{pt}}}\right\}
.\end{align*}
By taking pushouts inductively we can realize
\begin{align*}
{\mathbb{A}}^n\setminus\left\{{{\operatorname{pt}}}\right\} \simeq{\Sigma}_{S^1} ({\mathbb{A}}^1\setminus\left\{{{\operatorname{pt}}}\right\}) \wedge({\mathbb{A}}^n\setminus\left\{{{\operatorname{pt}}}\right\}) \simeq(S^1){ {}^{ \scriptscriptstyle\wedge^{n-1} }  } \wedge({\mathbb{G}}_m){ {}^{ \scriptscriptstyle\wedge^{n} }  }
.\end{align*}

\end{remark}

\begin{remark}

We can use this to write
\begin{align*}
{\mathbb{P}}^n/{\mathbb{P}}^{n-1} 
&\simeq{{\mathbb{P}}^n \over {\mathbb{P}}^n\setminus\left\{{{\operatorname{pt}}}\right\} }\\
&\simeq{{\mathbb{A}}^n \over {\mathbb{A}}^n\setminus\left\{{{\operatorname{pt}}}\right\} }\\
&\simeq{{\operatorname{pt}}\over {\mathbb{A}}^n\setminus\left\{{{\operatorname{pt}}}\right\} }\\
&\simeq{\Sigma}_{S^1} \qty{ {\mathbb{A}}^n\setminus\left\{{{\operatorname{pt}}}\right\} } \\
&\simeq(S^1){ {}^{ \scriptscriptstyle\wedge^{n} }  } \wedge({\mathbb{G}}_m){ {}^{ \scriptscriptstyle\wedge^{n} }  }
.\end{align*}

\end{remark}

\begin{remark}

Stable homotopy shows that inverting \({\Sigma}\) is useful, which we
also do in the \({\mathbb{A}}^1{\hbox{-}}\)setting by inverting
\({\Sigma}_{{\mathbb{P}}^1} ({-}) \coloneqq{\mathbb{P}}^1 \wedge({-})\)
to obtain a stable homotopy category \({\mathsf{SH}}(k)\).

\end{remark}

\begin{theorem}[Morel, Hopkins-Morel]

For \(k\in \mathsf{Field}\), stably we have
\begin{align*}
[S^0, S^0] \xrightarrow{\sim} [{\mathbb{P}}^n/{\mathbb{P}}^{n-1}, {\mathbb{P}}^n/{\mathbb{P}}^{n-1}] \xrightarrow{\sim} {\operatorname{GW}}(k)
.\end{align*}
Moreover, there is a ring structure on homotopy classes which yields an
isomorphism of rings into \textbf{Milnor-Witt \(K{\hbox{-}}\)theory},
\begin{align*}
\bigoplus_{n\in {\mathbb{Z}}} [ S^0, {\mathbb{G}}_m{ {}^{ \scriptscriptstyle\wedge^{n} }  }] \xrightarrow{\sim}  {\mathsf{K}}^{\scriptscriptstyle \mathrm{MW}} _*(k)
.\end{align*}

\end{theorem}

\begin{remark}

\({\mathsf{K}}^{\scriptscriptstyle \mathrm{MW}} _*\) is a graded
associative algebra with generators
\([u] \in {\mathsf{K}}^{\scriptscriptstyle \mathrm{MW}} _1(k)\) for
\(u\in k^{\times}\) and
\(\eta \in {\mathsf{K}}^{\scriptscriptstyle \mathrm{MW}} _{-1}(k)\),
with relations

\begin{itemize}
\tightlist
\item
  \([u][1-u] = 0\), the \emph{Steinberg relations},
\item
  \([ab] = [a] + [b] + \eta[a][b]\),
\item
  \([a]\eta = \eta[a]\),
\item
  \(\eta q_{\operatorname{hyp}}= 0\) for
  \(q_{\operatorname{hyp}}\coloneqq\eta[-1] + 2\)
\end{itemize}

\end{remark}

\begin{remark}

There is an isomorphism
\begin{align*}
{\operatorname{GW}}(k) &\xrightarrow{\sim}  {\mathsf{K}}^{\scriptscriptstyle \mathrm{MW}} _0(k) \\
\left\langle{a}\right\rangle &\rightleftharpoons 1 + \eta[a] \\
q_{\operatorname{hyp}}\coloneqq\left\langle{1}\right\rangle + \left\langle{-1}\right\rangle &\rightleftharpoons 1 + 1 + \eta[-1]
.\end{align*}

\end{remark}

\begin{remark}[on the proof]

\([a]\) yields a map
\begin{align*}
[a]: S^0 = (\operatorname{Spec}k)^{\coprod 2} &\to {\mathbb{G}}_m \\
p &\mapsto a
,\end{align*}
where \(p\) is the non-basepoint, and
\begin{align*}
\eta: {\mathbb{A}}^2\setminus\left\{{{\operatorname{pt}}}\right\} &\to {\mathbb{P}}^1 \\
(x, y) &\mapsto [x: y]
.\end{align*}
On \({\mathbb{C}}{\hbox{-}}\)points,
\({\mathbb{C}}^2\setminus\left\{{0}\right\}\simeq S^3\) maps to
\({\mathbb{CP}}^1\simeq S^2\) by the Hopf map, but on
\({\mathbb{R}}{\hbox{-}}\)points we get
\(S^1 \xrightarrow{\deg = -2} S^1\) implying that \(\eta\) is not
nilpotent, which is a new fact.

\end{remark}

\begin{remark}

We can define
\begin{align*}
X\vee Y = {X \times Y \over {\operatorname{pt}}_X \sim {\operatorname{pt}}_Y}
\end{align*}
and get maps
\begin{align*}
X\vee Y \to X\times Y\to X\wedge Y
.\end{align*}
which yields
\begin{align*}
{\Sigma}(X\times Y) \xrightarrow{\sim } {\Sigma}X \vee{\Sigma}Y \vee{\Sigma}(X\wedge Y)
.\end{align*}

\end{remark}

\begin{lemma}[?]

In \({\mathsf{SH}}(k)\),we get

\begin{center}
\begin{tikzcd}
    {{\mathbb{G}}_m{ {}^{ \scriptscriptstyle\times^{2} }  } } && {{\mathbb{G}}_m} \\
    \\
    { {{\mathbb{G}}_m{ {}^{ \scriptscriptstyle\wedge^{2} }  }} \vee{{\mathbb{G}}_m{ {}^{ \scriptscriptstyle\wedge^{2} }  }} } && {{\mathbb{G}}_m}
    \arrow["{\operatorname{mult}}", from=1-1, to=1-3]
    \arrow["{(1, 1, \eta)}", from=3-1, to=3-3]
    \arrow["\simeq", from=1-1, to=3-1]
    \arrow[equal, from=1-3, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXEdHX21ee1xcdGltZXMgMn0iXSxbMCwyLCJcXEdHX21ee1xcd2VkZ2Vwcm9kIDJ9IFxcd2VkZ2Vwcm9kIFxcR0dfbV57XFxzbWFzaHByb2QgMn0iXSxbMiwyLCJcXEdHX20iXSxbMiwwLCJcXEdHX20iXSxbMCwzLCJcXG11bHQiXSxbMSwyLCIoMSwgMSwgXFxldGEpIl0sWzAsMSwiXFxzaW1lcSIsMV0sWzMsMiwiXFxzaW1lcSIsMV1d}{Link
to Diagram}
\end{quote}

\end{lemma}

\begin{lemma}[?]

The map
\begin{align*}
f: {\mathbb{P}}^1 &\to {\mathbb{P}}^1 \\
z &\mapsto az
\end{align*}
is equal to \(1 + \eta[a]\) in \({\mathsf{SH}}(k)\), since
\(f = {\Sigma}g\) where
\begin{align*}
g: {\mathbb{G}}_m &\to {\mathbb{G}}_m \\ 
z &\mapsto az
,\end{align*}
which is equal to
\begin{align*}
{\Sigma}({\mathbb{G}}_m \times k \xrightarrow{1\times a} {\mathbb{G}}_m^{\times 2} \xrightarrow{{\operatorname{mult}}} {\mathbb{G}}_m)
.\end{align*}

\end{lemma}

\begin{remark}

The lemma implies the relation \([ab] = [a] + [b] + \eta[a][b]\), and it
turns out there's an isomorphism to motivic homotopy groups of spheres:
\begin{align*}
 {\mathsf{K}}^{\scriptscriptstyle \mathrm{MW}} _*(k) \xrightarrow{\sim} \bigoplus _{n\in {\mathbb{Z}}} [S^0, {\mathbb{G}}_m{ {}^{ \scriptscriptstyle\wedge^{n} }  }]
.\end{align*}

\end{remark}

\hypertarget{big-problems}{%
\subsection{Big Problems}\label{big-problems}}

\begin{remark}

Notation: we'll write
\begin{align*} 
\bigoplus_{n\in {\mathbb{Z}}} \pi_{n, n} {\mathbb{S}}\coloneqq\bigoplus _{n\in {\mathbb{Z}}} [S^0, {\mathbb{G}}_m{ {}^{ \scriptscriptstyle\wedge^{n} }  } ]
\end{align*}
to be the zero line of the homotopy groups of spheres, and generally
\(\bigoplus _n \pi_{n+r, n} {\mathbb{S}}\) for the \(r{\hbox{-}}\)line.
Classical homotopy groups of spheres encode interesting geometric
information, and we're finding that the corresponding motivic homotopy
groups of spheres do as well.
\href{https://arxiv.org/abs/1604.00365}{Röndigs-Spitzweck-Østvær}
compute the 1-line for \(\operatorname{ch}k\neq 2\) in a 2019 Annals
paper, and we have some information about the 2-line.

\end{remark}

\begin{question}

What is
\([{\mathbb{P}}^n/{\mathbb{P}}^{n-1}, {\mathbb{P}}^n/{\mathbb{P}}^{n-1}]\)
for more general rings?
\href{https://arxiv.org/abs/2102.01618}{Bachmann-Østvær (2021)} do this
over \({\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{2} } \right] }\) and
show
\begin{align*}
\pi_{0,0}{\mathbb{S}}\otimes{ {\mathbb{Z}}_{\widehat{2}} } \xrightarrow{\sim} {\operatorname{GW}}({\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{2} } \right] })\otimes{ {\mathbb{Z}}_{\widehat{2}} }
.\end{align*}

\end{question}

\begin{question}

What is \(\pi_{*, *}{\mathbb{S}}\) in general?

\end{question}

\begin{question}

Is there a Freudenthal suspension theorem? I.e. which stable elements of
\(\pi_{*, *} {\mathbb{S}}\) correspond to unstable groups?

\end{question}

\hypertarget{counting-things}{%
\subsection{Counting Things}\label{counting-things}}

\begin{remark}

Many people have used the \({\mathbb{A}}^1{\hbox{-}}\)Euler class for
interesting things! Let \(X\in {\mathsf{sm}}{\mathsf{Sch}}_{/ {k}}\)
with \(\dim X = d\) and let
\(V\to X \in { { {\mathsf{Bun}}_{\operatorname{GL}_r} }}_{/ {X}}\) be a
vector bundle.

\end{remark}

\begin{definition}[Orientation of bundles]

A bundle \(V\to X\) is \textbf{oriented} by the following data:
\((L, \phi)\) where \(L\to X\) is a line bundle and \(\phi\) is a
trivialization
\begin{align*}
\phi: \operatorname{det}V { { \, \xrightarrow{\sim}\, }}L^{\otimes 2}
.\end{align*}
It is \textbf{relatively oriented} when
\(\mathop{\mathrm{Hom}}(\operatorname{det}TX, \operatorname{det}V)\) is
oriented, where
\(\operatorname{det}({-}) = \bigwedge\nolimits^{\text{top}}({-})\).

\end{definition}

\begin{example}[?]

For \(X = {\mathbb{P}}^n\) or \({\operatorname{Gr}}(m, n)\)
(parameterizing copies of \({\mathbb{P}}^m\) in \({\mathbb{P}}^n\)),
then
\begin{align*}
\omega_x = \operatorname{det}T^n X = {\mathcal{O}}(-n-1)
,\end{align*}
and \(X\) is orientable iff \(n\) is odd. For \({\mathbb{P}}^1\),
\({\mathcal{O}}(n)\) is relatively orientable iff \(n\) is even.

\end{example}

\hypertarget{euler-numbers}{%
\subsection{Euler Numbers}\label{euler-numbers}}

\begin{definition}[Euler Number in $\GW(k)$]

Suppose \(X\in { {\mathsf{sm}}{\mathsf{Sch}}}_{/ {k}}\) is proper with
\(\dim X = d\) and consider a vector bundle

\begin{center}
\begin{tikzcd}
\operatorname{GL}_d \curvearrowright F
  \ar[r] 
& 
V
  \ar[d] 
\\
& 
X 
\ar[u, "f", bend left, dotted]
\end{tikzcd}
\end{center}

Suppose

\begin{itemize}
\tightlist
\item
  \(V\) is relatively oriented by \((L, \phi)\), and \(f\) is a section
  with isolated zeros, so \(\left\{{f=0}\right\}\) consists of zeros of
  multiplicity 1, or equivalently
\item
  For all \(x\in \left\{{f = 0}\right\}\), the composite map
  \begin{align*}
  Tf \coloneqq\qty{T_x X \to T_{f(x)} V \xrightarrow{\sim} T_x X \oplus V_x \xrightarrow{p_2} V_x} && f(x) \coloneqq(x, 0)
  \end{align*}
  has nonvanishing determinant.
\end{itemize}

Then the \textbf{Euler number of \((V, \phi)\)} with respect to \(f\) is
defined as
\begin{align*}
n(V, \phi, f) \coloneqq\sum_{x \in \left\{{f = 0 }\right\} \subseteq X} \deg_x f
.\end{align*}
where \(\deg_x f\) can be computed by

\begin{itemize}
\tightlist
\item
  Choosing local Nisnevich coordinates on \(X\) ,
\item
  Choosing local trivializations of \(V\) which are ``compatible'' with
  \(\phi\),
\end{itemize}

Then locally writing
\begin{align*}
f: {\mathbb{A}}^d \to {\mathbb{A}}^d \implies J_f \coloneqq\operatorname{det}\qty{{\partial}f_i \over {\partial}x_j}
,\end{align*}
one has
\begin{align*}
\text{For } J_f(x) \neq 0 \in \kappa(x), \quad
\deg_x f \coloneqq\operatorname{Tr}_{\kappa(x) _{/ {k}} } \left\langle{J_f(x)}\right\rangle
.\end{align*}

\end{definition}

\begin{remark}

Equivalently, \(T_x f\in \mathop{\mathrm{Hom}}(T_x X, V_x)\) and we can
define
\begin{align*}
J_f(x) \coloneqq\operatorname{det}T_x f \in \mathop{\mathrm{Hom}}(\operatorname{det}T_x X, \operatorname{det}V_x) { { \, \xrightarrow{\sim}\, }}L_x{ {}^{ \scriptstyle\otimes_{k}^{2} }  }
,\end{align*}
where the orientation provides the isomorphism. Picking any basis for
\(L_x{ {}^{ \scriptstyle\otimes_{k}^{2} } }\) yields a number which is
well-defined in \(\kappa(x) / (\kappa(x)^{\times})^2\) by choosing a
trivialization of \(L_x\).

\end{remark}

\begin{question}

What happens if the zeros of \(f\) have multiplicity \(m_i > 1\)? In the
classical setting, we didn't say what happens when \(J_f(x) = 0\). We'll
answer this next time.

\end{question}

\begin{question}

Why is the Euler number \(n(V, f)\) independent of the section \(f\)?
Analogously, why is the number of intersections in the original problem
2, not depending on which specific lines were chosen?

\end{question}

\begin{answer}

Sections with isolated zeros are often connected by 1-parameter
\({\mathbb{A}}^1{\hbox{-}}\)families of such sections, and
\({\operatorname{GW}}(k[x]) \xrightarrow{\sim} {\operatorname{GW}}(k)\),
although this is hard to show.

Alternatively, the Euler number is a pushforward of an Euler class
taking values in interesting cohomology theories, so
\(n(V, f) = \pi_* e(V, f)\).

\end{answer}

\hypertarget{kirsten-wickelgren-talk-2-friday-july-16}{%
\section{Kirsten Wickelgren, Talk 2 (Friday, July
16)}\label{kirsten-wickelgren-talk-2-friday-july-16}}

\hypertarget{intro-1}{%
\subsection{Intro}\label{intro-1}}

\begin{remark}

Recall that we have a classical degree map
\begin{align*}
\deg: [S^n, S^n] \to {\mathbb{Z}}
\end{align*}
which roughly counts preimages. Given
\(f\in \mathop{\mathrm{Hom}}_{\mathsf{Top}}(S^n, S^n)\) and
\(p\in S^n\), we write
\(f^{-1}(p) = \left\{{ { {q}_1, {q}_2, \cdots, {q}_{N}} }\right\}\) and
have a formula \(\deg f = \sum \deg_{q_i} f\) where the local degrees
\(\deg_{q_i}f\) can be computed by picking orientation-compatible
coordinates \({ \left\{{ {x}_1, {x}_2, \cdots, {x}_{n} }\right\} }\)
near \(q_i\) and
\({ \left\{{ {y}_1, {y}_2, \cdots, {y}_{n} }\right\} }\) near \(p\). In
these coordinates we can form the Jacobian
\(J_f \coloneqq\operatorname{det}{\frac{\partial f_i}{\partial x_j}\,}\)
and write
\begin{align*}
\deg_x f =
\begin{cases}
+1 & J(q_i) > 0 
\\
-1 & J(q_i) < 0.
\end{cases}
\end{align*}

\end{remark}

\begin{question}

What happens if the zeros of \(f\) are not of multiplicity 1, so
\(J_f(q_i) = 0\)?

\end{question}

\begin{proposition}[Eisenbud–H.Levine–Khimshiashvili signature formula]

Over \(k\coloneqq{\mathbb{R}}\), any quadratic form can be diagonalized
to \(\operatorname{diag}(1,\cdots,1, -1,\cdots,-1)\), and there is a
formula
\begin{align*}
\deg_x f = \operatorname{sig}\omega^{{\mathrm{EKL}}}
\end{align*}
where \(\omega^{{\mathrm{EKL}}}\) is the isomorphism class of the
bilinear form defined in the following way: for
\(f = (f_1, \cdots, f_n)\), set
\begin{align*}
Q \coloneqq{{\mathbb{R}}[x_1, \cdots, x_n]_0 \over \left\langle{{ {f}_1, {f}_2, \cdots, {f}_{n}} }\right\rangle }
\end{align*}
which is a finite dimensional local complete intersection. Since \(Q\)
is Gorenstein\footnote{The dualizing sheaf is locally free}, there is an
isomorphism \(\mathop{\mathrm{Hom}}_k(Q, k) \xrightarrow{\sim} Q\),
which we can take to be the bilinear form.\footnote{Even better, there
  is a distinguished isomorphism coming from a distinguished socle
  element (Scheja-Storch).}

\end{proposition}

\begin{remark}

This form can be made very explicit: writing
\(J_f = \operatorname{det}{\frac{\partial f_i}{\partial x_j}\,} \in Q\),
choose a \(k{\hbox{-}}\)linear map \(\eta: Q\to k\) such that
\(\eta(J_f) = \dim_k Q\) and set
\begin{align*}
\omega^{{\mathrm{EKL}}} &\coloneqq\qty{ Q{ {}^{ \scriptstyle\otimes_{k}^{2} }  } \xrightarrow{{\operatorname{mult}}} Q \xrightarrow{\eta} k} \\ \\
\implies \omega^{{\mathrm{EKL}}}: Q{ {}^{ \scriptstyle\otimes_{}^{2} }  } &\to k \\
(g, h) &\mapsto \eta(gh)
.\end{align*}
It turns out that the isomorphism class of \(\omega^{{\mathrm{EKL}}}\)
does not depend on the choice of \(\eta\).

\end{remark}

\begin{example}[?]

Let \(f:{\mathbb{A}}^1\to {\mathbb{A}}^1\) and \(f(z) = z^2\) with
\(q=0\). Then
\begin{align*}
Q = {k[x]_0 \over \left\langle{x^1}\right\rangle } \xrightarrow{\sim} {k[x]\over \left\langle{x^2}\right\rangle}
\end{align*}
and \(J_f = 2x\). We then get
\(\omega^{{\mathrm{EKL}}} = { \begin{bmatrix}  {0} & {1} \\  {1} & {0} \end{bmatrix} }\),
which up to a change of basis is
\(h \coloneqq{ \begin{bmatrix}  {1} & {0} \\  {0} & {-1} \end{bmatrix} }\).

\end{example}

\begin{question}

Eisenbud notes that \(\omega^{\mathrm{EKL}}\) is defined over fields of
arbitrary characteristic not equal to 2, does it have a topological
interpretation?

\end{question}

\begin{remark}

Yes! It comes from the \({\mathbb{A}}^1{\hbox{-}}\)degree.

\end{remark}

\begin{theorem}[Kass-W.]

\(\omega^{{\mathrm{EKL}}} = \deg_q^{{\mathbb{A}}^1}f\) is the local
degree in \({\operatorname{GW}}(k)\) when \(\kappa(q) = k\).
\href{https://arxiv.org/pdf/1912.04788.pdf}{Brazelton, Burklund, Mckean,
Montoro, Opie} handle the case when \(\kappa(q)/k\) is separable.

\end{theorem}

\hypertarget{mathbba1hbox-milnor-numbers}{%
\subsection{\texorpdfstring{\({\mathbb{A}}^1{\hbox{-}}\)Milnor
numbers}{\{\textbackslash mathbb\{A\}\}\^{}1\{\textbackslash hbox\{-\}\}Milnor numbers}}\label{mathbba1hbox-milnor-numbers}}

\begin{definition}[Node, hypersurface singularity]

For \(\operatorname{ch}k\neq 2\), the simplest type of singularity is a
\textbf{node}, defined over
\(\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\) as a
point \(p\in X\) such that\footnote{Her \({\mathcal{O}}_{X, p}\) is the
  stalk of the structure sheaf at \(p\), which is a local ring with a
  unique maximal ideal \({\mathfrak{m}}_p\), and the LHS is completion
  at that ideal, so
  \begin{align*}
  \widehat{{\mathcal{O}}_{X, p}} \coloneqq\qty{{\mathcal{O}}_{X, p}}{ {}_{ \widehat{{\mathfrak{m}}_p} } }
  .\end{align*}
}
\begin{align*}
\widehat{{\mathcal{O}}_{X, p}}
\xrightarrow{\sim} 
{ \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu  { \left[ { { {x}_1, {x}_2, \cdots, {x}_{n}} } \right] }  \over \sum x_i^2 + \text{h.o.t.} }
,\end{align*}

A \textbf{hypersurface singularity} is a point
\(p\in \left\{{f=0}\right\} \subseteq X\).

\end{definition}

\begin{definition}[Milnor number]

Let \(k\coloneqq{\mathbb{C}}\). If you vary \(X\) in a family
\begin{align*}
X_+ \coloneqq\left\{{f(x_1,\cdots, x_n) + \sum a_i x_i = t}\right\}
,\end{align*}
then the singularity \(p\) bifurcates into nodes. The number of nodes is
given by the \textbf{Milnor number}, defined as \(M_p\), the number of
nodes in the family \(X_+\) for any sufficiently small
\(\left\{{a_i}\right\}\). For \(R={\mathbb{C}}\), this is explicitly
described as
\begin{align*}
M_p \coloneqq\deg_p^{{\mathsf{Top}}}\operatorname{grad}f
.\end{align*}

\end{definition}

\begin{remark}

For other \(k\) with \(\operatorname{ch}k \neq 2\), nodes come in
different types: given a residue field \(L\) at a node \(p\), the
tangent directions defined over some extension \(L[\sqrt a]\) for
\(a\in L^{\times}/ (L^{\times}){ {}^{ \scriptscriptstyle\times^{2} } }\).

\end{remark}

\begin{example}[?]

Over \(k={\mathbb{R}}\), one has examples like

\begin{itemize}
\item
  \(x_1^2 +x_2^2 = 0\), yielding a non-split node and non-rational
  tangent directions

  \includegraphics{figures/2021-07-19_19-58-25.png}
\item
  \(x_1^2-x_2^2=0\), yielding a split node and rational tangent
  directions

  \includegraphics{figures/2021-07-16_13-26-04.png}
\end{itemize}

\end{example}

\begin{definition}[Type of a node]

Let \(p\) be a node with
\begin{align*}
\widehat{{\mathcal{O}}_{X, p}} { { \, \xrightarrow{\sim}\, }}{ L { \left[\left[ { { {x}_1, {x}_2, \cdots, {x}_{n}} } \right] \right] }  \over \sum a_i x_i^2 }, 
&& L \coloneqq\kappa(p)
.\end{align*}
The \textbf{type} \(p\) is defined as
\begin{align*}
{\operatorname{type}}(p) \coloneqq\operatorname{Tr}_{L_{/ {k}} } \left\langle{2^n \prod_{i=1}^n a_i}\right\rangle \in {\operatorname{GW}}(k)
.\end{align*}

\end{definition}

\begin{definition}[$\AA^1\dash$Milnor numbers]

The \textbf{\({\mathbb{A}}^1{\hbox{-}}\)Milnor number} is defined as
\begin{align*}
M_p \coloneqq\deg_p^{{\mathbb{A}}_1} \operatorname{grad}f = \sum_{p\in N} {\operatorname{type}}(p)
,\end{align*}
\(N\) is the set of nodes of \(f\) in a family for a generic
\(\left\{{{ {a}_1, {a}_2, \cdots, {a}_{n}} }\right\}\).

\begin{quote}
Note: the second equality is due to Kass-Wickelgren.
\end{quote}

\end{definition}

\begin{example}[?]

Let \(f(x,y)\coloneqq x^3-y^2\) with \(\operatorname{ch}k\neq 2,3\),
then

\begin{itemize}
\tightlist
\item
  \(p=(0, 0) \in \left\{{f=0}\right\}\),
\item
  \(\nabla f = (3x^2, -2y)\), and
  \begin{align*}
  \deg^{{\mathbb{A}}^1} \nabla f 
  &= \deg^{{\mathbb{A}}^1} (x\mapsto 3x^2 ) \deg^{{\mathbb{A}}^1} (y\mapsto -2y) \\
  &= {
  \begin{bmatrix}
    {0} & {1/3} 
  \\
    {1/3} & {0}
  \end{bmatrix}
  } \left\langle{-2}\right\rangle \\
  &= \left\langle{1}\right\rangle + \left\langle{-1}\right\rangle \\
  &= q_{{\operatorname{hyp}}}
  ,\end{align*}
  which has rank 2 and thus \(M = 2\). This yields a cusp.
\end{itemize}

The family \(y^2 = x^3 + ax + t\) for \(a\neq 0\) yields a family:

\includegraphics{figures/2021-07-16_13-33-13.png}

\begin{itemize}
\tightlist
\item
  In the first term, the cusp bifurcates into 2 nodes, yields
  \(\operatorname{rank}M^{{\mathbb{A}}^1}(C) = M(C)\) for \(C\) the cusp
  curve
\item
  In the second, there are singular fibers when \(x^3+ax+t\) has double
  roots, which happens when the discriminant is zero, so this occurs iff
  \(-4a^3 -27b^2 =0\).
\end{itemize}

This yields a bifurcation into two nodes, and for example,

\begin{itemize}
\tightlist
\item
  Over \({\mathbb{F}}_5\):
  \(\left\langle{1}\right\rangle = \left\langle{-1}\right\rangle\), so
  this can not bifurcate into 1 split and 1 non-split rational node.
\item
  Over \({\mathbb{F}}_7\):
  \(\left\langle{1}\right\rangle \neq \left\langle{-1}\right\rangle\),
  so this can not bifurcate into 2 split or 2 non-split nodes.
\end{itemize}

\end{example}

\begin{remark}

The classical Milnor number appears in conductor formulas and are
related to the Euler characteristic \(\chi\) of the \emph{Milnor fiber}.
M. Levine-Lehalleur-Srinivas and R. Azouri have subtle quadratic
enrichments to \({\operatorname{GW}}(k)\).

\end{remark}

\hypertarget{mathbba1hbox-euler-characteristic-chimathbba1}{%
\subsection{\texorpdfstring{\({\mathbb{A}}^1{\hbox{-}}\)Euler
characteristic
\(\chi^{{\mathbb{A}}^1}\)}{\{\textbackslash mathbb\{A\}\}\^{}1\{\textbackslash hbox\{-\}\}Euler characteristic \textbackslash chi\^{}\{\{\textbackslash mathbb\{A\}\}\^{}1\}}}\label{mathbba1hbox-euler-characteristic-chimathbba1}}

\begin{definition}[Relatively oriented bundles]

For
\(X\in {\mathsf{sm}}{\mathsf{Var}}^{\mathop{\mathrm{proj}}}_{/ {k}}\),
we saw last time that a vector bundle \(V\to X\) is \textbf{relatively
oriented} by \((L, \phi)\) where \(L\to X\) is a line bundle and
\begin{align*}
\phi: L{ {}^{ \scriptstyle\otimes_{k}^{2} }  } \xrightarrow{\sim} \mathop{\mathrm{Hom}}(\operatorname{det}{\mathsf{T}}X, \operatorname{det}V)
.\end{align*}

\end{definition}

\begin{remark}

The tangent bundle \({\mathsf{T}}X\) has a canonical relative
orientation since
\begin{align*}
\mathop{\mathrm{Hom}}(\operatorname{det}{\mathsf{T}}X, \operatorname{det}{\mathsf{T}}X) \xrightarrow{\sim} {\mathcal{O}}\xrightarrow{\sim} {\mathcal{O}}{ {}^{ \scriptstyle\otimes_{}^{2} }  }
\end{align*}
where \({\mathcal{O}}\) is a trivial bundle of rank 1. It follows that
we may define the Euler number of tangent bundle:
\begin{align*}
\chi^{{\mathbb{A}}^1}(X) \coloneqq n({\mathsf{T}}X) \in {\operatorname{GW}}(k)
,\end{align*}
where \(n\) is the Euler number.

\end{remark}

\begin{theorem}[M. Levine]

\(\chi^{{\mathbb{A}}^1}(X)\) is equal to the \textbf{categorical Euler
characteristic}. The definition is omitted here, see
\href{https://arxiv.org/abs/1703.03049}{Levine's ``Enumerative geometry
with quadratic forms''}.

\end{theorem}

\begin{example}[M. Levine-Lehalleur-Srinivas]

\envlist

\begin{quote}
See \url{https://arxiv.org/abs/2101.00482}.
\end{quote}

Let \(X \subseteq {\mathbb{P}}^{n+1}\) for \(n\) even, written
\(X \coloneqq\left\{{F=0}\right\}\) for
\(f\in {\mathbb{R}}[x_0, \cdots, x_n]_{e}\) homogeneous of degree \(e\).
Take
\begin{align*}
\omega^{{\mathrm{EKL}}}: Q{ {}^{ \scriptstyle\otimes_{}^{2} }  }\to k && 
Q \coloneqq{k[x_0,\cdots, x_n] \over \left\langle{ {\frac{\partial f}{\partial x_0}\,}, \cdots, {\frac{\partial f}{\partial x_n}\,} }\right\rangle }
.\end{align*}

Define \(B_{\operatorname{Jac}}\) to be the restriction of
\(\omega^{\mathrm{EKL}}\) to \(\bigoplus_{q=0}^n Q_{(q+1)e-n-2}\), then
\begin{align*}
\chi^{{\mathbb{A}}^1}(X) = \left\langle{e}\right\rangle + \left\langle{-e}\right\rangle B_{\operatorname{Jac}} + {n\over 2}q_{{\operatorname{hyp}}}
.\end{align*}

\end{example}

\begin{example}[Clebsch surface]

Take the Clebsch surface
\begin{align*}
C \coloneqq\left\{{ {\left[ {{ {x}_1, {x}_2, \cdots, {x}_{3}}} \right]} {~\mathrel{\Big|}~}\sum x_i^3 = \qty{\sum x_i}^3  }\right\} \subseteq {\mathbb{P}}^3_{/ {{\mathbb{Q}}}} 
.\end{align*}
Then \(\chi(C) = 9\), and
\begin{align*}
\chi^{{\mathbb{A}}^1}(C) = 2q_{{\operatorname{hyp}}} + \left\langle{-10}\right\rangle + \left\langle{-6}\right\rangle + \left\langle{-21}\right\rangle + \left\langle{-14}\right\rangle + \left\langle{-2}\right\rangle
,\end{align*}
which can be computed with Macaulay2!

\end{example}

\hypertarget{cohomology-and-the-euler-class}{%
\subsection{Cohomology and the Euler
Class}\label{cohomology-and-the-euler-class}}

\begin{remark}

Note that we have a stable \({\mathbb{A}}^1{\hbox{-}}\)homotopy category
\({\mathsf{SH}}(k)\), so we can take cohomology theories on
\(X\in {\mathsf{sm}}{\mathsf{Sch}}_{/ {k}}\):

\begin{itemize}
\tightlist
\item
  \(H_{ \mathrm{mot}}\) or \(H{\mathbb{Z}}\): motivic cohomology
\item
  \(\tilde H_{ \mathrm{mot}}\) or \(H\tilde {\mathbb{Z}}\): extended
  motivic cohomology
\item
  \({\mathsf{K}}\): \({\mathsf{K}}{\hbox{-}}\)theory
\item
  \({\operatorname{KO}}\): Hermitian \({\mathsf{K}}{\hbox{-}}\)theory
\end{itemize}

It turns out that
\begin{align*}
H^n(X) = \pi_{-n}\mathop{\mathrm{Hom}}_{{\mathsf{SH}}(k)}(X, H) = [X, {\Sigma}^n H]
,\end{align*}
and it's useful to allow twisting the second term by shifts.

\end{remark}

\begin{definition}[Twisted suspension]

Letting \(V\to X\) be a vector bundle, we can define the \textbf{twisted
suspension}
\begin{align*}
H^V(X) \coloneqq[X, \mathop{\mathrm{Th}}(V) \wedge H]
&& \text{where } 
\mathop{\mathrm{Th}}(V) \coloneqq{V {}^{ \vee }\over V {}^{ \vee }\setminus\left\{{0}\right\}} = {{\mathbb{P}}(V {}^{ \vee }\oplus {\mathcal{O}}) \over {\mathbb{P}}(V {}^{ \vee })}
\end{align*}
where taking the dual is a condition needed for \(K{\hbox{-}}\)theory.
This reduces to a usual suspension for the trivial bundle.

\end{definition}

\begin{example}[?]

For
\(V\coloneqq({\mathcal{O}}_X){ {}^{ \scriptscriptstyle\oplus^{n} } }\)
the trivial rank \(n\) bundle on \(X\), we have \(H^V = H^n\).

\end{example}

\begin{example}[?]

\begin{align*}
H{\mathbb{Z}}^n(X) \cong H_{ \mathrm{mot}} ^{2n}(X; {\mathbb{Z}}(n)) \cong H^{2n, n}_{ \mathrm{mot}} (X) \cong {\operatorname{CH}}^n(X)
,\end{align*}
the Chow group of codimension \(n\) cycles modulo rational equivalence.
These are the \emph{geometric gradings}.

\end{example}

\begin{example}[?]

\(\tilde{H{\mathbb{Z}}}^n(X) \cong \tilde{{\operatorname{CH}}}^n(X)\)
the \textbf{Chow-Witt group}, also called the \textbf{oriented Chow
group}. Using the Gersten resolution, these can be expressed as formal
sums of codimension \(n\) subvarieties with coefficients in
\({\operatorname{GW}}(k(Z))\), where \(k(Z)\) are the rational functions
of \(Z\), subject to some conditions, modulo equivalence.

\end{example}

\begin{example}[?]

\({\mathsf{K}}^0(X)\) is the group completion of vector bundles on
\(X\), and \({\operatorname{KO}}^0(X)\) additionally remembers the data
of a nondegenerate symmetric bilinear form. These are ``representable''
in a sense because they are related to Thom spectra.

\end{example}

\begin{definition}[Cohomology with support]

Let \(Z\hookrightarrow X\) be a closed subscheme, then define the
\textbf{cohomology with support}
\begin{align*}
H_Z^V(X) \coloneqq { \left[ { { X \over X\setminus Z}, \mathop{\mathrm{Th}}(V) \wedge H} \right] } _{{\mathsf{SH}}(k)}
.\end{align*}

\end{definition}

\begin{definition}[Euler class]

For \(H\) a cohomology theory and \({\mathbb{S}}|to H\) a ring? and a
vector bundle \(V\to X\) with a section \(f\) (for example, the zero
section), then the \textbf{Euler class}
\begin{align*}
e^H(V, f) \in H_{\left\{{f=0}\right\}}^{V {}^{ \vee }}(X)
\end{align*}
is the class of the map
\begin{align*}
{X\over X\setminus\left\{{f=0}\right\}} \xrightarrow{f} {V\over V\setminus\left\{{0}\right\}} \wedge H
.\end{align*}

\end{definition}

\begin{definition}[Euler Number]

For \(f:X\to S\) is a \textbf{local closed immersion} (\(\mathrm{lci}\))
if it locally factors as \(U \xrightarrow{i} P \xrightarrow{p} S\) with
\(p\) smooth and \(i\) a closed immersion determined by a \emph{Koszul
regular sequence} (so modding out doesn't yield a zero divisor, and the
higher cohomology of the Koszul complex is zero).

\end{definition}

\begin{remark}

Some properties:

\begin{itemize}
\item
  This has a well-behaved \textbf{cotangent complex} \(L_f\).
\item
  For a regular embedding, \(L_i \simeq N_u {}^{ \vee }P[1]\) is
  \textbf{conormal bundle}.\footnote{I.e. the dual of the normal bundle.}
\item
  \(L_p \simeq\Omega_{P/S} \simeq{\mathsf{T}}_p {}^{ \vee }\).
\item
  \(L_{p_i}\) is determined by \(i^* L_p\to L_{p_i} \to L_i\),
\item
  There is a coherent Serre duality related to \(L_f\).
\end{itemize}

There is also a good notion of pushforward: let \(p:X\to S\) be proper
\footnote{Here \textbf{proper} means that the preimage of a compact set
  is compact.} and \(\mathrm{lci}\).

Then there is a \textbf{Becker-Gottlieb transfer}
\begin{align*}
\operatorname{Tr}^{\text{BG}}: {\Sigma}_+^\infty S\to \mathop{\mathrm{Th}}(L_p)
,\end{align*}
and the cartoon is the following:

\includegraphics{figures/2021-07-16_14-06-20.png}

Here we embed \(X\to S\) into the trivial bundle over \(S\) and take a
neighborhood. Letting \(B\) be the trivial bundle over \(S\), then
\(\mathop{\mathrm{Th}}(B) \cong \Sigma_+^\infty S\) and collapsing
fiberwise quotients by the complement of the neighborhood:

\includegraphics{figures/2021-07-16_14-08-44.png}

This yields \(p_*: H^{L_p}(X) \to H^0(S)\).

\end{remark}

\hypertarget{oriented-cohomology-theories-and-euler-numbers}{%
\subsection{Oriented Cohomology Theories and Euler
Numbers}\label{oriented-cohomology-theories-and-euler-numbers}}

\begin{remark}

\envlist

\begin{itemize}
\item
  \(H\) is \(\operatorname{GL}{\hbox{-}}\)oriented if
  \(H_Z^n(X) \xrightarrow{\sim} H_Z^V(X)\) with
  \(n\coloneqq\operatorname{rank}V\).

  An example is \(H{\mathbb{Z}}, {\mathsf{K}}\), but non examples are
  \({H\tilde {\mathbb{Z}}}, {\operatorname{KO}}\).
\item
  \(H\) is \({\operatorname{SL}}{\hbox{-}}\)oriented if

  \begin{itemize}
  \tightlist
  \item
    \(H_Z^V(X) \xrightarrow{\sim} H_Z^{V'}(X)\),
  \item
    \(\operatorname{rank}V = \operatorname{rank}V'\) and
  \item
    \(\operatorname{det}V \xrightarrow{\sim} \operatorname{det}V' \otimes L{ {}^{ \scriptstyle\otimes_{}^{2} } }\)
    for \(L\to X\) a line bundle.
  \end{itemize}

  An example is \(H\tilde{{\mathbb{Z}}}, {\operatorname{KO}}\).
\end{itemize}

\end{remark}

\begin{remark}

For \(V\to X\) a relatively oriented vector bundle on
\(X \xrightarrow{p} k\) with \(p\) smooth and proper, and \(H\) an
\({\operatorname{SL}}_c{\hbox{-}}\)oriented cohomology theory. Then
\begin{align*}
H^{V {}^{ \vee }}(X) \xrightarrow{\sim} H^{{\mathsf{T}} {}^{ \vee }X}(X)
.\end{align*}
Letting \(f\) be any section of \(V\), e.g.~the zero section, then

\begin{center}
\begin{tikzcd}
    {e^H(V, f) \in} & {H_{\left\{{f=0}\right\}}^{V {}^{ \vee }}(X)} &&& {H^{V {}^{ \vee }}(X)} \\
    & {X\to {X\over X\setminus\left\{{f=0}\right\}}} &&& {\mathop{\mathrm{Th}}(V) \wedge H} \\
    & {}
    \arrow["{z: \text{ forget support}}", from=1-2, to=1-5]
    \arrow[maps to, from=2-2, to=2-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNixbMSwwLCJIX3tcXHRze2Y9MH19XntWXFxkdWFsfShYKSJdLFs0LDAsIkhee1ZcXGR1YWx9KFgpIl0sWzEsMl0sWzAsMCwiZV5IKFYsIGYpIFxcaW4iXSxbMSwxLCJYXFx0byB7WFxcb3ZlciBYXFxzbXRze2Y9MH19Il0sWzQsMSwiXFxUaChWKSBcXHNtYXNocHJvZCBIIl0sWzAsMSwiejogXFx0ZXh0eyBmb3JnZXQgc3VwcG9ydH0iXSxbNCw1LCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJtYXBzIHRvIn19fV1d}{Link
to Diagram}
\end{quote}

Letting \(f: X\to \mathop{\mathrm{Th}}(V) \wedge H\), any two sections
\(f_1, f_2\) of \(V\) are connected by copies of \({\mathbb{A}}^1\) in
\(H^0(V)\), so
\begin{align*}
e^H(V) \coloneqq z(e^H(V, f_1)) = z(e^H(V, f_2))
.\end{align*}

\end{remark}

\begin{definition}[Euler number]

The \textbf{Euler number} \(n^H(V)\) of \(V\) in \(H^0(S)\) is
\begin{align*}
n^H(V) \coloneqq p_* e^H(V)
.\end{align*}

\end{definition}

\begin{remark}

This agrees with
\begin{align*}
n(V, f) = \sum_{x\in \left\{{f= 0 }\right\} \subseteq X} \deg_x f, && \deg_x f \in {\operatorname{GW}}(k)
,\end{align*}
for \(H = H\tilde{{\mathbb{Z}}}, {\operatorname{KO}}\) over
\(S\coloneqq\operatorname{Spec}k\). Moreover, \(n(V, f)\) is independent
of the choice of section.

\begin{quote}
See \href{https://arxiv.org/abs/1805.05920}{Déglise-Jin-Khan},
\href{https://arxiv.org/abs/2002.01848}{Bachmann-Wickelgren}.
\end{quote}

\end{remark}

\hypertarget{arithmetic-count-of-lines-on-a-smooth-cubic-surface-joint-with-jesse-kass}{%
\subsection{Arithmetic count of lines on a smooth cubic surface (joint
with Jesse
Kass)}\label{arithmetic-count-of-lines-on-a-smooth-cubic-surface-joint-with-jesse-kass}}

\begin{definition}[Cubic surface]

A \textbf{cubic surface} is given by \(X \subseteq {\mathbb{P}}^3\)
where \(X = \left\{{f=0}\right\}\) with \(f\in k[w,x,y,z]_3\)
homogeneous of degree 3.

\end{definition}

\begin{theorem}[Salmon-Cayley, 1849]

Any smooth cubic surface over \({\mathbb{C}}\) contains exactly 27
lines.

\end{theorem}

\begin{example}[?]

Consider the Fermat cubic:
\begin{align*}
f(x,y,z,w) = x^2 + y^2 + z^2 + w^2
.\end{align*}
The lines are given by
\begin{align*} 
\left\{{ {\left[ {S, -S, T, -T} \right]}  {~\mathrel{\Big|}~}[S, T] \in {\mathbb{P}}^1({\mathbb{C}}) \subseteq X }\right\}
.\end{align*}
For \(\omega^3,\lambda^3 = 1\), we can take
\begin{align*} 
\left\{{{\left[ {S, \lambda S, T, \omega T} \right]} {~\mathrel{\Big|}~}[S, T]\in {\mathbb{P}}^1({\mathbb{C}}) }\right\} 
.\end{align*}
Permuting the variables in \({4\choose 2}/2 =3\) ways, then there are
\(3\cdot 3 \cdot 3 = 27\) total lines.

\end{example}

\begin{proof}[?]

Let \({\operatorname{Gr}}(1, 3)\) be the Grassmannian parameterizing
\(W \subseteq {\mathbb{C}}^4\) with \(\dim W=2\), or equivalently copies
of \({\mathbb{P}}^1\) in \({\mathbb{P}}^3\). Take the tautological
\(S\to {\operatorname{Gr}}(1, 3)\) whose fiber over \([{\mathbb{P}}W]\)
is \(W\) itself. Then
\(\operatorname{Sym}^3 S {}^{ \vee }_{[{\mathbb{P}}W]} = \operatorname{Sym}^3(W {}^{ \vee })\)
are cubic polynomials in \(W\), and \(f\) determines a section
\(\sigma_f\) of \(\operatorname{Sym}^3 S {}^{ \vee }\) given by
\begin{align*}
\sigma_f([{\mathbb{P}}W) = { \left.{{f}} \right|_{{W}} }
,\end{align*}
which is zero iff the line \({\mathbb{P}}W\) is contained in \(X\). So
we've reduced to counting zeros of a section:
\begin{align*}
n(V) = \sum_{\text{lines } L \subseteq X} \deg_L \sigma_f
.\end{align*}

\begin{fact}

For a smooth cubic surface, all zeroes of \(\sigma_f\) have multiplicity
one.

\end{fact}

Over \({\mathbb{C}}\), the classical differential topological degree is
1, so \(n(V)\) is the number of lines, and \(n(V) = 27\).

Consider now the situation over \({\mathbb{R}}\):

\begin{itemize}
\tightlist
\item
  Schläfli (1861): Over \({\mathbb{R}}\), there can only be
  \(3, 7, 15\), or \(27\) lines.
\item
  Segre (1942): These lines break into two classes: hyperbolic, or
  elliptic.
\end{itemize}

For \(L \subseteq X\) a real line, \(L\) gives an involution
\(I:L\to L\) where \(I(p)\) is defined by consider \(T_p X\), which
contains a line \(L\) along with a curve \(C\) of degree 2 since the
total degree is 3:

\includegraphics{figures/2021-07-16_14-28-57.png}

In particular, there is another point in the intersection of
\(L \cap C \cap X\), so \(I(p)\) is defined as that point where
\(T_{I(p)}X = T_pX\). If \(I\) yields a
\({\mathbb{C}}{\hbox{-}}\)conjugate pair, say \(L\) is elliptic, and
otherwise if \(I\) yields two \({\mathbb{R}}{\hbox{-}}\)point say \(L\)
is hyperbolic. Taking a path \(p\to I(p)\), if the tangent space
undergoes a full twist, this is the elliptic situation.

\end{proof}

\begin{theorem}[Segre + many authors in the 2010s]

The number of hyperbolic lines minus the number of elliptic lines is
exactly 3.

\end{theorem}

\begin{question}

What about other fields, like
\(k = {\mathbb{F}}_p, { {\mathbb{Q}}_p }, {\mathbb{Q}}\)?

\end{question}

\begin{answer}

The above proof works in \({\mathbb{A}}^1{\hbox{-}}\)homotopy theory.
Letting \(X \subseteq X \subseteq {\mathbb{P}}^3_{/ {k}}\) a cubic
surface.

\end{answer}

\begin{definition}[Type of a line]

The \textbf{type} of a line \(L\) is an element
\begin{align*}
\left\langle{D}\right\rangle \in {\operatorname{GW}}(k(L)),
&&
D \in k(L)^{\times}/ (k(L)^{\times}){ {}^{ \scriptscriptstyle\times^{2} }  }
\end{align*}
such that the fixed points \(\mathrm{Fix} (I)\) form a conjugate pair of
points defined over \(k(L) { \left[ {\sqrt D} \right] }\).

\end{definition}

\begin{theorem}[Kass-Wickelgren]

let \(k\in \mathsf{Field}^{\operatorname{ch}\neq 2}\) and
\(X \subseteq {\mathbb{P}}^3_{/ {k}}\), then
\begin{align*}
\sum_{\substack{\text{lines} \\ L \subseteq k(X) }} \operatorname{Tr}_{k(L) / k} {\operatorname{type}}(D) = 15\left\langle{1}\right\rangle + 12\left\langle{-1}\right\rangle \in {\operatorname{GW}}(k)
.\end{align*}

\end{theorem}


\printbibliography[title=Bibliography]


\end{document}
