Sbornik: Mathematics, 2025, Volume 216, Issue 4, Pages 578–580 DOI: https://doi.org/10.4213/sm10194e (Mi sm10194)

DOI: https://doi.org/10.4213/sm10194e

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In the introduction of the authors’ paper [6] it was erroneously stated that extremal curve germs of types $(\mathrm{ID}^\vee)$ and $(\mathrm{IE}^\vee)$ are exceptional, that is, their general anticanonical member $D\in |-K_X|$ is not Du Val of type $\mathrm A$. The correct statement should be as follows: in the case $(\mathrm{IE}^\vee)$ $D$ is always of type $\mathrm{A}_{7}$ and in the case $(\mathrm{ID}^\vee)$ $D$ is of type $\mathrm{A}_{3}$ or $\mathrm{D}_{k}$, $k\geqslant 4$, depending on whether the non-Gorenstein point $P\in X$ is of type $\mathrm{cA}/2$ or $\mathrm{cAx}/2$ (see [1], (1.2.4)). This mistake does not affect any assertions in [6] nor their proofs. However, for the convenience of references, we provide below the classification of irreducible $\mathbb{Q}$-conic bundle germs which contain extremal curve germs of type $(\mathrm{ID}^\vee)$ and $(\mathrm{IE}^\vee)$ and their general anticanonical members.

Notation. Let $(X,C)$ be a $\mathbb{Q}$-conic bundle germ with irreducible $C$, let $f\colon X\to Z$ be the corresponding contraction, and let $D\in |-K_X|$ be the general member. If $D\supset C$, then $\Delta(D,C)$ denotes the dual graph of the minimal resolution $\mu\colon \widetilde D\to D$. For this graph we use the standard notation: each vertex $\circ$ corresponds to a prime $\mu$-exceptional divisor (which is a $(-2)$-curve on $\widetilde D$) and the vertex $\bullet$ corresponds to the proper transform of $C$ (which is a $(-1)$-curve).

The cases with singular $Z$ (see [1])

Case $(\mathrm{T})$; see [1], (1.2.1). The divisor $D$ does not contain $C$, and it is a disjoint union $D=D_1+D_2$, where $D_1\simeq D_2$ is a singularity of type $\mathrm{A}_{m-1}$, $m\geqslant 2$.

Case $(\mathrm{k2A})$; see [1], (1.2.2) and Theorem 11.1. The inclusion $D\supset C$ holds, where $\Delta(D,C)$ is as follows:

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Case $(\mathrm{IE}^\vee)$; see [1], (1.2.3). The intersection $D\cap C$ is a single point $P$, and the singularity $(D, P)$ is of type $\mathrm{A}_{7}$.

Case $(\mathrm{ID}^\vee)$; see [1], (1.2.4). The intersection $D\cap C$ is a point $\{P\}$, and $(D, P)$ is of type $\mathrm{A}_{3}$ or $\mathrm{D}_{k}$, where $k\geqslant 4$.

Case $(\mathrm{IA}^\vee)$; see [1], (1.2.5). The intersection $D\cap C$ is a point $\{P\}$, and $(D, P)$ is of type $\mathrm{A}_{3}$.

Case $(\mathrm{II}^\vee)$; see [1], (1.2.6). The intersection $D\cap C$ is a point $\{P\}$, and $(D, P)$ is of type $\mathrm{D}_{2k+1}$, $k\geqslant 2$.

The cases with smooth $Z$ (see [1]–[4])

The Gorenstein case. The linear system $|-K_X|$ is base point free, and $(D, P)$ is smooth.

Case $(\mathrm{cAx}/2)$; see [1], § 12, and [3], § 7. The intersection $D\cap C$ is a point $\{P\}$, and $(D, P)$ is of type $\mathrm{D}_{k}$, $k\geqslant 4$.

Case $(\mathrm{cD}/2)$; see [1], § 12, and [3], § 7. The intersection $D\cap C$ is a point $\{P\}$, and $(D, P)$ is of type $\mathrm{D}_{2k}$, $k\geqslant 3$.

Case $(\mathrm{cE}/2)$; see [1], § 12, and [3], § 7. The intersection $D\cap C$ is a point $\{P\}$, and $(D, P)$ is of type $\mathrm{E}_{7}$.

Case $(\mathrm{k1A})$; see [1], Theorem 8.6(iii). The intersection $D\cap C$ is a point $\{P\}$, and $(D, P)$ is of type $\mathrm{A}_{k}$, $k\geqslant 1$.

Case $(\mathrm{cD/3})$; see [1], Theorem 8.6(iii). The intersection $D\cap C$ is a point $\{P\}$, and $(D, P)$ is of type $\mathrm{E}_{6}$.

Case $(\mathrm{IC})$; see [2], Theorem 1.3, and [4], Theorem 1.1. The inclusion $D\supset C$ holds, the unique non-Gorenstein point has index $5$, and $\Delta(D,C)$ is as follows:

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Case $(\mathrm{IIA})$; see [5]. The intersection $D\cap C$ is a point $\{P\}$, and $(D, P)$ is of type $\mathrm{D}_{2k+1}$, $k\geqslant 2$.

Case $(\mathrm{IIB})$; see [2], Theorem 1.3. The inclusion $D\supset C$ holds, and $\Delta(D,C)$ is as follows:

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Case $(\mathrm{kAD})$; see [2], Theorem 1.3. The inclusion $D\supset C$ holds, and $\Delta(D,C)$ is as follows:

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Case $(\mathrm{k3A})$; see [2], Theorem 1.3. The inclusion $D\supset C$ holds, and $\Delta(D,C)$ is as follows:

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Bibliography
1.	S. Mori and Yu. Prokhorov, “On $\mathbb Q$-conic bundles”, Publ. Res. Inst. Math. Sci., 44:2 (2008), 315–369
2.	S. Mori and Yu. Prokhorov, “On $\mathbb Q$-conic bundles. III”, Publ. Res. Inst. Math. Sci., 45:3 (2009), 787–810
3.	S. Mori and Yu. Prokhorov, “Threefold extremal contractions of type (IA)”, Kyoto J. Math., 51:2 (2011), 393–438
4.	S. Mori and Yu. Prokhorov, “3-fold extremal contractions of types (IC) and (IIB)”, Proc. Edinb. Math. Soc. (2), 57:1 (2014), 231–252
5.	S. Mori and Yu. G. Prokhorov, “Threefold extremal contractions of type (IIA). I”, Izv. Math., 80:5 (2016), 884–909
6.	S. Mori and Yu. G. Prokhorov, “General elephants for threefold extremal contractions with one-dimensional fibres: exceptional case”, Sb. Math., 212:3 (2021), 351–373

Citation in format AMSBIB

\Bibitem{MorPro25}
\by Sh.~Mori, Yu.~G.~Prokhorov
\paper Remark on the paper ``General elephants for threefold extremal contractions with one-dimensional fibres: exceptional case'', Sbornik: Mathematics, 212:3 (2021), 351--373
\jour Sb. Math.
\yr 2025
\vol 216
\issue 4
\pages 578--580
\mathnet{http://mi.mathnet.ru/eng/sm10194}
\crossref{https://doi.org/10.4213/sm10194e}
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