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.

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.

Reviewer: Jing Zhang (University Park)