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\begin{document}

Report on the paper "Multiple fibers of del Pezzo fibrations"\ by
S.\,Mori and Yu.\,Prokhorov.

Let $\pi\colon X\to Z$ be a flat morphism with connected fibers
such that $Z$ is a smooth curve, and $X$ is a threefold with
terminal singularities. Let $F$ be a scheme fiber of $\pi$ over a
point $P\in Z$. Suppose that $-K_{X}$ is $\pi$-nef and $\pi$-big.
Suppose, in addition, that $F$ is a multiple fiber of $\pi$, i.e.
$F=mF_{0}$ for $m\geqslant 2$, where $F_{0}$ is reduced at some
point. The main result of the paper "Multiple fibers of del Pezzo
fibrations"\ says that $m\leqslant 6$ and all cases $m=2,\ldots,6$
occur. Moreover, the paper also describes the possible
singularities of $X$ along $F_{0}$.

One case of the considered problem is of special importance.
Namely, if we assume that $X$ has $\mathbb{Q}$-factorial
singularities and $\mathrm{rk}\,\mathrm{Pic}(X/Z)=1$, then $\pi$
is a del Pezzo fibration that is a Mori fibred space. Note that
for a given rationally connected threefold, we have three possible
outcomes for Minimal Model Program (Mori Program): a terminal
$\mathbb{Q}$-factorial Fano threefold of Picard rank $1$, a
standard conic bundle, and a del Pezzo fibration that is a Mori
fibred space. Thus, to analyze a possible values of $m$ and
possible singularities that can occur along $F$ is of big
importance for birational geometry.

The answer for singularities of $X$ along $F$ is almost complete.
Namely, the authors obtained a possible list of "baskets of
singularities"\ at which $F_{0}$ is not a Cartier divisor. And
almost all (except possibly few cases) do occur. The paper
contains plenty of nice and very explicit examples (which goes
quite well with a style of the authors).

As I already said, the main result is very important for
birational geometry. The idea of the proof is very elegant and
surprisingly simple. It is based on the orbifold Riemann-Roch
formula for $lF_{0}$, standard vanishing theorems and a subtle
usage of Minimal Model Program, where $l\in\mathbb{N}$. However,
some part of the proof is quite technical. The exposition is very
clear and easy to read. The language is good. Without any doubts,
the paper deserves to be published in your journal.

I have few comments on the paper that I think can help readers to
understand the paper better. My comments have nothing to do with
the mathematical content of the paper.

\begin{enumerate}
\item The introduction is very short. Many people do not know what
is "basket of singularities". It seems possible to add few lines
explaining what "baskets"\ are.

\item In the main theorem, it would be better to mention that "the
type"\ of the fiber $F$ is basically the corresponding basket of
singularities.

\item The introduction lacks examples. But the paper contains
plenty of explicit examples (it has a special section for
examples). May be it would be better to put one (or few) example
in the introduction.

\item The main result is also related to a problem of fiberwise
birational transformation of del Pezzo fibrations (considered by
Corti, Grinenko, Park and Pukhlikov). This problem is strongly
related to singularities of $F$ and $X$ along $F$. So the case
when $F$ is multiple (i.e. $m\geqslant 2$) is considered to be the
case of "very bad singularities", which must lead to some
non-trivial birational fiberwise transformation of the del Pezzo
fibration $\pi$ along $F$. It would be worth while to mention
this.

\end{enumerate}


\bigskip

Sincerely, Ivan Cheltsov









\end{document}
