Speaker
Description
The topic is finite diagonal subgroups $A\subset$SL$(4,\mathbb{C})$ and their \hbox{$A$-Hilbert} schemes. As a dimension reducing assumption, I impose the additional $(1,2)$-symmetric condition. The case to bear in mind is $\frac1r(1,1,a,b)$ with $r = a+b+2$. The ``junior end and all-even'' conditions for the quotient $X=\mathbb{A}^4/A$ to have a crepant resolution are known from Sarah Davis's thesis [D].
Studying the $A$-Hilbert scheme $A$-Hilb$\mathbb{A}^4$ in the general $(1,2)$-symmetric case is interesting in its own right, and provides more detailed insight into case of the crepant resolution. The variety $Y=A$-Hilb$\mathbb{A}^4$ is toric, a union of affine pieces corresponding to monomomial ideals $I\subset k[\mathbb{A}^4]=k[x,y,z,t]$, and can be constructed by my 2009 computer algebra routine [M]. In very many cases $Y$ is nonsingular, and is a resolution $Y\to X$ with exceptional divisors of discrepancy 0 or 1.
The calculation of $A$-Hilb$\mathbb{A}^4$ mirrors the classical construction of Nakamura [A] and Craw--Reid [CR], with some remarkable modifications.
