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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2312.14271 |
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| _version_ | 1866910079944491008 |
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| author | Wahl, Jonathan |
| author_facet | Wahl, Jonathan |
| contents | A surface pair $(X,C)$ is a germ of a normal surface singularity $(X,0)$ and a sum $C=\sum c_iC_i$ 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 $(Σ, γ_i,n_i)$. There are natural notions of morphism and log cover of surface pairs. We introduce a volume $Vol(X,C)$ in $\mathbb Q_{\geq 0}$, 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 $f:(X',C')\rightarrow (X,C)$ is a morphism of degree $d$, then $Vol(X',C')\geq d\cdot Vol(X,C)$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_14271 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Volume of a Surface or Orbifold Pair Wahl, Jonathan Algebraic Geometry 32S50 (Primary), 14J17, 57M10 (Secondary) A surface pair $(X,C)$ is a germ of a normal surface singularity $(X,0)$ and a sum $C=\sum c_iC_i$ 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 $(Σ, γ_i,n_i)$. There are natural notions of morphism and log cover of surface pairs. We introduce a volume $Vol(X,C)$ in $\mathbb Q_{\geq 0}$, 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 $f:(X',C')\rightarrow (X,C)$ is a morphism of degree $d$, then $Vol(X',C')\geq d\cdot Vol(X,C)$, 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. |
| title | The Volume of a Surface or Orbifold Pair |
| topic | Algebraic Geometry 32S50 (Primary), 14J17, 57M10 (Secondary) |
| url | https://arxiv.org/abs/2312.14271 |