diff --git a/main.tex b/main.tex index 9bc843685b57fa9fb0b0a6dabe7dfa9548f45f3d..b91585cabaa8b1831008313fade6f08c87779eda 100644 --- a/main.tex +++ b/main.tex @@ -277,10 +277,13 @@ $\nu_{\alpha,\beta}(u)=0$, and hence $\chern_2^{\alpha, \beta}(u)=0$. In fact, this allows us to use the characteristic curves of some $v$ and $u$ (with $\Delta(v), \Delta(u)\geq 0$ and positive ranks) to determine the location of the pseudo-wall where $u$ pseudo-semistabilizes $v$. This is done by finding the -intersection of $\Theta_v$ and $\Theta_u$. -%TODO ref forwards +intersection of $\Theta_v$ and $\Theta_u$, the point $(\beta,\alpha)$ where +$\nu_{\alpha,\beta}(u)=\nu_{\alpha,\beta}(v)=0$, and a pseudo-wall point on +$\Theta_v$, and hence the apex of the circular pseudo-wall with centre $(\beta,0)$ +(as per subsection \ref{subsect:bertrams-nested-walls}). \subsection{Bertram's nested wall theorem} +\label{subsect:bertrams-nested-walls} Although Bertram's nested wall theorem can be proved more directly, it's also important for the content of this document to understand the connection with @@ -625,10 +628,11 @@ rank that appears turns out to be $\sage{extravagant.actual_rmax}$. \section{B.Schmidt's Wall Finding Method} -The proof for the previous theorem was hinted at in [ref], but the value appears -explicitly in [ref]. The latter reference is a SageMath [ref?] library for -computing certain quantities related to Bridgeland stabilities on Picard rank 1 -varieties. It also includes functions to compute pseudo-walls and +The proof for the previous theorem was hinted at in +\cite{SchmidtBenjamin2020Bsot}, but the value appears explicitly in +\cite{SchmidtGithub2020}. The latter reference is a SageMath \cite{sagemath} +library for computing certain quantities related to Bridgeland stabilities on +Picard rank 1 varieties. It also includes functions to compute pseudo-walls and pseudo-semistabilizers for tilt stability. Here is an outline of the algorithm involved to do this. Simplifications will be @@ -662,8 +666,8 @@ semistabilizers, as given by theorem \ref{thm:loose-bound-on-r}. Recalling consequence 2 of lemma \ref{lem:pseudo_wall_numerical_tests}, we can iterate through the possible values of $\mu(u)=\frac{c}{r}$ taking a decreasing sequence of all fractions between $\mu(v)$ and $\beta_{-}$, who's denominators -are no large than $r_{max}$ (giving a finite sequence). This can be done with Farey sequences [ref], for -which there exist formulae to generate. +are no large than $r_{max}$ (giving a finite sequence). This can be done with +Farey sequences \cite[chapter 6]{alma994504533502466}, for which there exist formulae to generate. These $\mu(u)$ values determine pairs $r,c$ up to multiples, we can then take all multiples which satisy $0<r\leq r_{max}$. @@ -710,8 +714,8 @@ alternative algorithm which will later be described in this article [ref]. & $(3, 2\ell, -2)$ & $(3, 2\ell, -\frac{15}{2})$ \\ \hline - Computation time for earlier [ref] program & \sim 20s & >1hr \\ - Computation time for [ref] program & \sim 50ms & \sim 50ms \\ + \cite[\texttt{tilt.walls_left}]{SchmidtGithub2020} exec time & \sim 20s & >1hr \\ + \cite{NaylorRust2023} exec time & \sim 50ms & \sim 50ms \\ \hline \end{tabular} \end{center} @@ -744,8 +748,8 @@ u = Chern_Char(*var("r c d", domain="real")) Δ = lambda v: v.Q_tilt() \end{sagesilent} -Recall [ref] that $\chern_1^{\beta}$ has fixed bounds in terms of -$\chern(F)$, and so we can write: +Recall from eqn \ref{eqn-tilt-cat-cond} that $\chern_1^{\beta}$ has fixed +bounds in terms of $\chern(F)$, and so we can write: \begin{sagesilent} ts = stability.Tilt diff --git a/references.bib b/references.bib index 2b8a8bb15313982afd664b0545578f4379dbf0bf..78f44248d3c9cae41f7e1de6e92b2cc699887378 100644 --- a/references.bib +++ b/references.bib @@ -121,4 +121,16 @@ title = {{S}ageMath, the {S}age {M}athematics {S}oftware {S}ystem ({V}ersion 9.6.0)}, note = {{\tt https://www.sagemath.org}}, year = {2022} +} +@book{alma994504533502466, + author = {Niven, Ivan}, + address = {New York ;}, + booktitle = {An introduction to the theory of numbers}, + edition = {Second edition.}, + keywords = {Number theory}, + language = {eng}, + lccn = {66017623}, + publisher = {Wiley}, + title = {An introduction to the theory of numbers / Ivan Niven, Herbert S. Zuckerman.}, + year = {1966}, } \ No newline at end of file