Month: August 2019

Walls for G-Hilb via Reid’s recipe

Posted on by lisnbaa

I posted a new paper on the arXiv this week.

As I’ve discussed before, I’m interested in the birational geometry of minimal resolutions of 3-fold Gorenstein quotient singularities; i.e. quotients of \mathbb{C}^3 by finite subgroups of \text{SL}_3(\mathbb{C}).

I’ve had some ideas floating around since my time at Nagoya University in 2016 and, with a lot of great input from Alastair Craw, I distilled a selection of these ideas into a decent theorem with some promising consequences.

To summarise, as I’ve written about before, resolutions of \mathbb{C}^3/G correspond (not bijectively) to chambers in a big vector space \Theta, at least when G is abelian. There is a particular resolution that has been studied more than any of the others, called the G-Hilbert scheme G\text{-Hilb}. In general, the chamber structure inside \Theta is quite mysterious – for instance, as the singularities here are not symplectic there is no reason to expect that the walls define full hyperplanes instead of just cones. Using the mechanics of Reid’s recipe – which is a clever, representation-theoretic way of labelling the exceptional fibre inside G\text{-Hilb} – I produced explicit inequalities defining the chamber for G\text{-Hilb} and determined which of them are necessary, thus which define walls. The work of Craw-Ishii allows us to say exactly what wall-crossing behaviour happens at each of these walls, again using Reid’s recipe.

Hope it’s interesting to you! I’m hoping to publish some applications of this technology soon in joint work with Yukari Ito.

Cross-referencing named list items in LaTeX

Posted on by lisnbaa

I found myself needing to cross-reference a list item with a nonstandard item label in an enumerate environment but when I tried the usual \label & \ref pairing I discovered that it would generate a reference to the section / subsection the list was in, instead of a reference to the specific list item as I’d hoped.

After some searching and several unsuccessful attempts, the best solution I found was this via this blog post, which works perfectly for my situation. I include the code below with an example use case featuring two items labelled A1 and A2.

Hope it’s helpful to you too!

\makeatletter
\def\namedlabel#1#2{\begingroup
    #2%
    \def\@currentlabel{#2}%
    \phantomsection\label{#1}\endgroup
}
\makeatother

...

\begin{enumerate}
\item[\namedlabel{itm:anitem}{A1}] An item.
\item[\namedlabel{itm:anoitem}{A2}] Another item.
\end{enumerate}

I like \ref{itm:anitem} more than \ref{itm:anoitem}.