The Volume of a Surface or Orbifold Pair

Abstract

A surface pair $(X,C)$ is a germ of a normal surface singularity $(X,0)$ and a sum $C=\sum c_i{C}_i$ of curves on $X$, with $c_i\in [0,1]$. An orbifold pair has $c_i=1/n_i$, as intersecting with a small sphere gives a $3$-dimensional orbifold $(\hat{I}\pounds ,{\hat{I}}_i^3,n_i)$. There are natural notions of morphism and log cover of surface pairs. We introduce a volume $Vol(X,C)$ in ${\mathbb{Q}}_{\ge 0}$, computable from any log resolution, analogous to that in our 1990 JAMS paper when $C=0$. Denoting $\bar{C}=\sum (1-c_i)C_i$, one has $(X,C)$ log canonical iff $Vol(X,\bar{C})=0$. The main theorem (5.4-5.6) is that $Vol(X,C)$ is ``characteristic'': if $f:(X^{\prime },C^{\prime })\to (X,C)$ is a morphism of degree $d$, then $Vol(X^{\prime },C^{\prime })\ge d\cdot Vol(X,C)$, with equality if $f$ is a log cover. We prove (6.7) that $Vol(X,\sum (1/n_i)C_i)=0$ iff the associated orbifold has finite or solvable orbifold fundamental group, and these are classified. In (8.2) is proved a key case of the DCC Volumes Conjecture: The set $\{Vol(X,\sum (1/n_i)C_i)|X\text{RDP}\}$ satisfies the DCC, with minimum non-$0$ volume 1/3528.

24 pages, For 115AM Jaca Conference in June, 2023; typos and small errors corrected; Introduction expanded; alternative definition of Volume