Resolving quotient singularities III

To conclude this miniseries on resolutions of quotient singularities, I present the full toric picture of what is happening, including gory computations of the self-intersections of exceptional curves in terms of Jung-Hirzebruch continued fractions. The intricacy of the computations maybe conceals the conciseness of the toric lexicon: these three pages of computations are best summarised by the following picture for the quotient singularity \frac{1}{7}(1,5)


From it one can immediately read that its [minimal] resolution has three exceptional curves with discrepancies -1/7, -2/7, and -3/7. From the continued fraction expansion of \frac{7}{5}=[2,2,3] one gains the equivalent information that there are three exceptional curves with self-intersection -2,-2, and -3.


Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )


Connecting to %s