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Threefold extremal contractions of type (IIA). I. (English. Russian original) Zbl 1359.14037
Izv. Math. 80, No. 5, 884-909 (2016); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 80, No. 5, 77-102 (2016).
Let $$(X, D)$$ be a log pair of data with a normal variety $$X$$ and an effective Weil $$\mathbb{Q}$$-divisor $$D=\sum d_iD_i$$ on $$X$$ such that $$K_X+ D$$ is an $$\mathbb{Q}$$-Cartier divisor.
Let $$\mu: Y\rightarrow X$$ be a log resolution of $$(X, D)$$ and each $$E_i$$ is an exceptional divisors, then $K_Y+\mu_*^{-1}D= \mu^*(K_X+D) + \sum_{E_i} a(E_i, X, D) E_i.$
The discrepancy of $$(X, D)$$ is $\text{discrep}(X, D)= \inf _E\{ a(E, X, D): E \text{ is an exceptional divisor on } X\}.$
$$(X, D)$$ is terminal, if discrep$$(X, D)>0$$.
A contraction is a proper surjective morphism $$f: X\rightarrow Z$$ of normal varieties such that $$f_*{\mathcal{O}}_X= {\mathcal{O}}_Z$$.
Assume that $$(X, C)$$ is an analytic germ of a threefold $$X$$ with terminal singularities along with a reduced complete curve $$C$$. $$(X, C)$$ is called an extremal curve germ if there is a contraction $$f: (X, C) \rightarrow (Z, o)$$ such that $$C=f^{-1}(o)_{\mathrm{red}}$$ and $$-K_X$$ is $$f-$$ample. $$f$$ is flipping if its exceptional locus coincides with $$C$$, and divisorial if its exceptional locus is two-dimensional. $$(X, C)$$ is a $$\mathbb{Q}$$-conic bundle germ if $$f$$ is not birational (then $$Z$$ is a surface). Let $$|{\mathcal{O}}_X|_C$$ be the linear subsystem of $$|{\mathcal{O}}_X|$$ consisting of members containing $$C$$. The main theorem the authors prove in this paper is the following.
Main Theorem. Let $$(X, C)$$ be an extremal curve germ and let $$f: (X, C) \rightarrow (Z, o)$$ be the corresponding contraction. Assume that $$(X, C)$$ is not flipping and it has a point $$P$$ of type (IIA). Furthermore, assume that the general member $$H\in |{\mathcal{O}}_X|_C$$ is normal. Then $$H$$ has only rational singularities. Moreover, there are exactly four cases for the dual graph of $$(X, C)$$ and all the possibilities do occur.
To prove the main theorem, the authors first prove a technical theorem.
Let $$(X, C)$$ and $$f: (X, C) \rightarrow (Z, o)$$ be as above. Let $$(X^{\sharp}, P^{\sharp})\rightarrow (X, P)$$ be the index-one cover. Denote $l(P)={\text{len}}_P I^{\sharp(2)}/I^{\sharp 2},$ where $$I^\sharp$$ is the ideal defining $$C^\sharp$$ (the pull back of $$C$$ in $$X^\sharp$$) and $$J^{(n)}$$ is the symbolic $$n-$$th power of a prime ideal $$J$$.
Theorem. Let $$(X, C\simeq {\mathbb{P}}^1)$$ be an extremal curve germ and let $$f: (X, C) \rightarrow (Z, o)$$ be the corresponding contraction. For every $$n\geq 1$$, define an $${\mathcal{O}}_C$$-module ${\text{gr}}^n_C{\mathcal{O}}=I^{(n)}/I^{(n+1)}.$ Assume that $$H^0( {\text{gr}}^1_C{\mathcal{O}})\neq 0$$ and $$(X, C)$$ is not flipping and it has a point $$P$$ of type (IIA). Then the general member $$H\in |{\mathcal{O}}_X|_C$$ is normal and has only rational singularities. Moreover, there are exactly four cases for the dual graph of $$(X, C)$$ and all the possibilities do occur.

##### MSC:
 14J30 $$3$$-folds 14E30 Minimal model program (Mori theory, extremal rays) 14E05 Rational and birational maps
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