\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.59pt{\vbox to8.59pt{\pgfpicture\makeatletter\hbox{\hskip
4.29594pt\lower-2.55984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.09595pt}{1.7361pt}\pgfsys@curveto{4.09595pt}{3.99825pt}{2.26215pt}{5.83205pt}{0.0pt}{5.83205pt}\pgfsys@curveto{-2.26215pt}{5.83205pt}{-4.09595pt}{3.99825pt}{-4.09595pt}{1.7361pt}\pgfsys@curveto{-4.09595pt}{-0.52605pt}{-2.26215pt}{-2.35985pt}{0.0pt}{-2.35985pt}\pgfsys@curveto{2.26215pt}{-2.35985pt}{4.09595pt}{-0.52605pt}{4.09595pt}{1.7361pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.7361pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.3229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
6b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\mathchoice{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 4b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 4b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.93pt{\vbox to10.93pt{\pgfpicture\makeatletter\hbox{\hskip
5.46465pt\lower-3.0341pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.26465pt}{2.43054pt}\pgfsys@curveto{5.26465pt}{5.33815pt}{2.90761pt}{7.69519pt}{0.0pt}{7.69519pt}\pgfsys@curveto{-2.90761pt}{7.69519pt}{-5.26465pt}{5.33815pt}{-5.26465pt}{2.43054pt}\pgfsys@curveto{-5.26465pt}{-0.47707pt}{-2.90761pt}{-2.8341pt}{0.0pt}{-2.8341pt}\pgfsys@curveto{2.90761pt}{-2.8341pt}{5.26465pt}{-0.47707pt}{5.26465pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 4b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.59pt{\vbox to8.59pt{\pgfpicture\makeatletter\hbox{\hskip
4.29594pt\lower-2.55984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.09595pt}{1.7361pt}\pgfsys@curveto{4.09595pt}{3.99825pt}{2.26215pt}{5.83205pt}{0.0pt}{5.83205pt}\pgfsys@curveto{-2.26215pt}{5.83205pt}{-4.09595pt}{3.99825pt}{-4.09595pt}{1.7361pt}\pgfsys@curveto{-4.09595pt}{-0.52605pt}{-2.26215pt}{-2.35985pt}{0.0pt}{-2.35985pt}\pgfsys@curveto{2.26215pt}{-2.35985pt}{4.09595pt}{-0.52605pt}{4.09595pt}{1.7361pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.7361pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.3229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
4b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}-\frac{\frac{Pr[P=1]}{B}\sum_{p\in
supp(P)\setminus\\{1\\}}\frac{Pr[P=p]}{OPT(p)}p^{k}(1-p)^{t-k+1}}{(\sum
f^{p}(k,t,1))(\sum f^{p}(k,t,0))}&\mbox{if }t=k-1\end{array}\right.$

Thus we have $\mathchoice{\leavevmode\hbox to14.43pt{\vbox
to14.43pt{\pgfpicture\makeatletter\hbox{\hskip 7.21367pt\lower-3.74146pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 6b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 6b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.93pt{\vbox to10.93pt{\pgfpicture\makeatletter\hbox{\hskip
5.46465pt\lower-3.0341pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.26465pt}{2.43054pt}\pgfsys@curveto{5.26465pt}{5.33815pt}{2.90761pt}{7.69519pt}{0.0pt}{7.69519pt}\pgfsys@curveto{-2.90761pt}{7.69519pt}{-5.26465pt}{5.33815pt}{-5.26465pt}{2.43054pt}\pgfsys@curveto{-5.26465pt}{-0.47707pt}{-2.90761pt}{-2.8341pt}{0.0pt}{-2.8341pt}\pgfsys@curveto{2.90761pt}{-2.8341pt}{5.26465pt}{-0.47707pt}{5.26465pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 6b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.59pt{\vbox to8.59pt{\pgfpicture\makeatletter\hbox{\hskip
4.29594pt\lower-2.55984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.09595pt}{1.7361pt}\pgfsys@curveto{4.09595pt}{3.99825pt}{2.26215pt}{5.83205pt}{0.0pt}{5.83205pt}\pgfsys@curveto{-2.26215pt}{5.83205pt}{-4.09595pt}{3.99825pt}{-4.09595pt}{1.7361pt}\pgfsys@curveto{-4.09595pt}{-0.52605pt}{-2.26215pt}{-2.35985pt}{0.0pt}{-2.35985pt}\pgfsys@curveto{2.26215pt}{-2.35985pt}{4.09595pt}{-0.52605pt}{4.09595pt}{1.7361pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.7361pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.3229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
6b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\geq-\mathchoice{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 5b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
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} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 5b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.93pt{\vbox to10.93pt{\pgfpicture\makeatletter\hbox{\hskip
5.46465pt\lower-3.0341pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.26465pt}{2.43054pt}\pgfsys@curveto{5.26465pt}{5.33815pt}{2.90761pt}{7.69519pt}{0.0pt}{7.69519pt}\pgfsys@curveto{-2.90761pt}{7.69519pt}{-5.26465pt}{5.33815pt}{-5.26465pt}{2.43054pt}\pgfsys@curveto{-5.26465pt}{-0.47707pt}{-2.90761pt}{-2.8341pt}{0.0pt}{-2.8341pt}\pgfsys@curveto{2.90761pt}{-2.8341pt}{5.26465pt}{-0.47707pt}{5.26465pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 5b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.59pt{\vbox to8.59pt{\pgfpicture\makeatletter\hbox{\hskip
4.29594pt\lower-2.55984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.09595pt}{1.7361pt}\pgfsys@curveto{4.09595pt}{3.99825pt}{2.26215pt}{5.83205pt}{0.0pt}{5.83205pt}\pgfsys@curveto{-2.26215pt}{5.83205pt}{-4.09595pt}{3.99825pt}{-4.09595pt}{1.7361pt}\pgfsys@curveto{-4.09595pt}{-0.52605pt}{-2.26215pt}{-2.35985pt}{0.0pt}{-2.35985pt}\pgfsys@curveto{2.26215pt}{-2.35985pt}{4.09595pt}{-0.52605pt}{4.09595pt}{1.7361pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.7361pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.3229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
5b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$. Combining
$\textstyle 5$ and $\textstyle 6$, we get that

$\begin{split}u_{t}(k)-l_{t}(k)\geq\mathchoice{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 5b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 5b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.93pt{\vbox to10.93pt{\pgfpicture\makeatletter\hbox{\hskip
5.46465pt\lower-3.0341pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.26465pt}{2.43054pt}\pgfsys@curveto{5.26465pt}{5.33815pt}{2.90761pt}{7.69519pt}{0.0pt}{7.69519pt}\pgfsys@curveto{-2.90761pt}{7.69519pt}{-5.26465pt}{5.33815pt}{-5.26465pt}{2.43054pt}\pgfsys@curveto{-5.26465pt}{-0.47707pt}{-2.90761pt}{-2.8341pt}{0.0pt}{-2.8341pt}\pgfsys@curveto{2.90761pt}{-2.8341pt}{5.26465pt}{-0.47707pt}{5.26465pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 5b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.59pt{\vbox to8.59pt{\pgfpicture\makeatletter\hbox{\hskip
4.29594pt\lower-2.55984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.09595pt}{1.7361pt}\pgfsys@curveto{4.09595pt}{3.99825pt}{2.26215pt}{5.83205pt}{0.0pt}{5.83205pt}\pgfsys@curveto{-2.26215pt}{5.83205pt}{-4.09595pt}{3.99825pt}{-4.09595pt}{1.7361pt}\pgfsys@curveto{-4.09595pt}{-0.52605pt}{-2.26215pt}{-2.35985pt}{0.0pt}{-2.35985pt}\pgfsys@curveto{2.26215pt}{-2.35985pt}{4.09595pt}{-0.52605pt}{4.09595pt}{1.7361pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.7361pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.3229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
5b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\left[\sum\frac{f^{p}(k,t+1,1)c^{p}(k,t+1,1)}{\sum
f^{p}(k,t+1,1)}-\sum\frac{f^{p}(k,t+1,0)c^{p}(k,t+1,0)}{\sum
f^{p}(k,t+1,0)}\right]\\\ &\geq 0\end{split}$

where the second inequality follows from the fact that
$\mathchoice{\leavevmode\hbox to14.43pt{\vbox
to14.43pt{\pgfpicture\makeatletter\hbox{\hskip 7.21367pt\lower-3.74146pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 5b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 5b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.93pt{\vbox to10.93pt{\pgfpicture\makeatletter\hbox{\hskip
5.46465pt\lower-3.0341pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.26465pt}{2.43054pt}\pgfsys@curveto{5.26465pt}{5.33815pt}{2.90761pt}{7.69519pt}{0.0pt}{7.69519pt}\pgfsys@curveto{-2.90761pt}{7.69519pt}{-5.26465pt}{5.33815pt}{-5.26465pt}{2.43054pt}\pgfsys@curveto{-5.26465pt}{-0.47707pt}{-2.90761pt}{-2.8341pt}{0.0pt}{-2.8341pt}\pgfsys@curveto{2.90761pt}{-2.8341pt}{5.26465pt}{-0.47707pt}{5.26465pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 5b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.59pt{\vbox to8.59pt{\pgfpicture\makeatletter\hbox{\hskip
4.29594pt\lower-2.55984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.09595pt}{1.7361pt}\pgfsys@curveto{4.09595pt}{3.99825pt}{2.26215pt}{5.83205pt}{0.0pt}{5.83205pt}\pgfsys@curveto{-2.26215pt}{5.83205pt}{-4.09595pt}{3.99825pt}{-4.09595pt}{1.7361pt}\pgfsys@curveto{-4.09595pt}{-0.52605pt}{-2.26215pt}{-2.35985pt}{0.0pt}{-2.35985pt}\pgfsys@curveto{2.26215pt}{-2.35985pt}{4.09595pt}{-0.52605pt}{4.09595pt}{1.7361pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.7361pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.3229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
5b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\geq 0$ and the
inductive assumption $u_{t+1}(k)\geq l_{t+1}(k)$ at stage $(t+1)$. ∎

With these individual properties, we can now describe a property of the
optimal prior-independent algorithm.

######  Corollary F.2.

There exists an optimal prior-independent algorithm such that all
deterministic algorithms in its support are information-symmetric, do not stop
on seeing bad weather and satisfy the thresholding property.

###### Proof.

Observe that Backwards Induction gave us an algorithm in which two properties
were satisfied, 4.8 and F.1. For any mixed strategy that the adversary plays,
a best response of the agent is satisfies the above three properties. Applying
4.5 then leads to the Corollary. ∎

F.2 tells us that we can work with a reduced game where strategies of the
algorithm satisfy the described three properties. We can then show a property
of the worst-case distribution of the adversary in equilibrium.

######  Lemma F.3.

For any given $T$, for a fixed strategy of the agent (an algorithm) which does
not stop on seeing bad weather, any best response of the adversary is a
discrete distribution.

###### Proof.

4.7 tells us that for a deterministic algorithm $A$, and a fixed distribution
$p\in[\delta,1]$, the cost suffered by the algorithm $A(p)$ is a polynomial in
$p$ with degree at most $T$. This would also hold for randomized algorithms as
they are nothing but a convex combination over deterministic algorithms.
Recall that the expected cost of the algorithm for a distribution $p$, is

$A(p)=\sum_{\textbf{X}\in\\{0,1\\}^{T}}A(\textbf{X})p^{\sum_{i=1}^{T}X_{i}}(1-p)^{T-\sum_{i=1}^{T}X_{i}}$

where $A(\textbf{X})$ is the cost of the algorithm along a sampled sequence
$\textbf{X}=(X_{1},\dots,X_{T})$.

First note that since we are in a setting where the class of algorithms are
those which do not stop on seeing a bad day of weather, the cost of an
algorithm cannot be a constant as a function of $p$. This is because
$A(\\{0,\dots,0\\})=0$. The cost of any algorithm is a polynomial of degree at
least one.

Let the algorithms cost be of degree one, that is, $A(p)=\alpha p$ for some
$\alpha>0$. Now for $p\in[\delta,\frac{B-1}{T-1}]$, the utility of the
adversary is $\frac{\alpha}{T}$, while for
$p\in(\frac{B-1}{T-1},\min\left\\{\frac{B-1}{T-2},1\right\\}]$, the utility of
the adversary is $\frac{\alpha}{B+(1-p)(T-1)}>\frac{\alpha}{T}$, where the
utility was obtained by replacing $\text{OPT}_{p}(p)$. Observe that the
utility is strictly increasing in $p$ and thus is maximized at
$\min\left\\{\frac{B-1}{T-2},1\right\\}$. For each interval
$p\in\left(\frac{B-1}{T-k},\min\left\\{\frac{B-1}{T-k-1},1\right\\}\right]$,
where $k\in[\min\\{\left\lceil T-B\right\rceil,T-1\\}]$, the utility is a
ratio of two polynomials in $p$ of degree at most $T$. Thus the number of
local extrema of the utility function in each interval are $O(T)$, and thus
the number of global maximizers in each interval are $O(T)$. Since a necessary
condition of a best response is that the probabilities in the support must be
global maximizers in each interval, we have that any best response has at most
$O(T^{2})$ probabilities in its support, as there are $O(T)$ intervals.

Let the algorithms cost be of degree at least two. In that case, the utility
in the interval $p\in[\delta,\frac{B-1}{T-1}]$ is no longer a constant but of
degree at least one. The same argument of $O(T)$ local maximizers in each
interval follows and we have that any best response has at most $O(T^{2})$
probabilities in its support. ∎

######  Corollary F.4.

For a given $T$, the worst-case distribution in the prior-independent ski-
rental problem is a discrete distribution.


# Flexibility of Affine cones over Mukai fourfolds of genus $g\geq 7$

Michael Hoff<EMAIL_ADDRESS>and  Hoang Le Truong  Institute of
Mathematics, VAST, 18 Hoang Quoc Viet Road, 10307 Hanoi, Viet Nam  Thang Long
Institute of Mathematics and Applied Sciences, Hanoi, Vietnam
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract.

We show that the affine cones over a general Fano-Mukai fourfold of genus
$g=7$, $8$ and $9$ are flexible. Equivalently, there is an infinitely
transitive action of the special automorphism group on such affine cones. In
particular, any Mukai fourfold of genus $7,8$ and $9$ is
$\mathbb{A}^{2}$-cylindrical.

###### Key words and phrases:

Affine cone, Flexibility, Generic Flexibility, Fano variety, Cylinder,
Automorphism, Group action of the additive group

2020 Mathematics Subject Classification: 14R20, 14J45, 14J50, 14R05.
The first author was partially supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID
286237555 - TRR 195. The second author was partially supported by a fund of
Vietnam National Foundation for Science and Technology Development (NAFOSTED)
and VAST under grant number NVCC01.04/22-23

##  1\. Introduction

Studying affine varieties and their automorphism groups is an active area of
research. A remarkable property of an affine variety is the infinitely
transitive action of the special automorphism group on it. Let $X$ be an
affine variety over $\mathbb{C}$. A one-parameter subgroup $H$ of ${\rm
Aut}(X)$ isomorphic as an algebraic group to the additive group
$\mathbb{G}_{a}$ is called a unipotent one-parameter subgroup. We denote by
${\rm SAut}(X)$ the special automorphism group, that is, the subgroup of ${\rm
Aut}(X)$ generated by all the one-parameter unipotent subgroups. Following
[AFKKZ13], the variety $X$ is called flexible if ${\rm SAut}(X)$ acts highly
transitively on the smooth locus ${\rm reg}(X)$, that is, it acts
$m$-transitively for any natural number $m$.

In dimension $2$, del Pezzo surfaces of degree $\geq 4$ with their pluri-
anticanonical polarizations have flexible affine cones ([Per13, PaW16]). In
higher dimensions, the affine cones over flag varieties of dimension $\geq 2$
are flexible by [AZK12]. In [MPS18], it is shown that the affine cone over the
secant (tangential) variety of Segre-Veronese varieties are flexible.
Furthermore, there are several constructions producing new flexible varieties
from a given flexible one, e.g. via suspensions or open subsets with
complement of codimension $\geq 2$ (see [AFKKZ13, AZK12, FKZ16, KaZ99]).
Furthermore, several families of examples of Fano varieties of dimension $3$
and $4$ and Picard number one admitting flexible affine cones have been
constructed [AZK12, APS14, MPS18, PrZ18, PrZ20], but a complete classification
is still far from being known.

In this paper we focus on the case in which $X$ is a Fano–Mukai fourfold,
namely a four dimensional complex smooth Fano variety of index $2$. Given a
smooth Fano fourfold $X$ with Picard rank $1$, the index of $X$ is the integer
$r$ such that $-K_{X}=rH$, where $H$ is the ample divisor generating the
Picard group: ${\mathrm{Pic}}(X)=\mathbb{Z}H$. The degree $d=\deg X$ is the
degree with respect to $H$. It is known that $1\leq r\leq 5$ and smooth Fano
fourfolds of index $r=2$ are called Mukai fourfolds; their degrees are even
and can be written as $d=2g-2$, where $g$ is called the genus of $X$.

Our main theorem is the following.

######  Theorem 1.1.

The affine cones over general Mukai fourfold of genus $g=7$, $8$ and $9$ are
flexible.

The only known flexible Mukai fourfolds are of genus $10$. According to
[PrZ18, PrZ20] a smooth del Pezzo fourfold $W_{5}\subset\mathbb{P}^{7}$ of
degree $5$ has flexible affine cones. Starting with the del Pezzo quintic
fourfold $W_{5}$, and performing suitable Sarkisov links, Prokhorov and
Zaidenberg showed that the affine cones over any Mukai fourfold of genus $10$
are flexible. We proceed in a similar fashion starting with $\mathbb{P}^{4}$,
but also use classical adjunction theory to prove our main theorem.

Furthermore, the proof of Theorem 1.1 makes substantial use of a geometric
property called $\mathbb{A}^{n}$-cylindricity, which is worthwhile to study
for its own sake and is closely related to the property of being flexible (see
[Per13, PrZ20]). Recall that a smooth projective variety $X$ over $\mathbb{C}$
is called $\mathbb{A}^{n}$-cylindrical if it contains an
$\mathbb{A}^{n}$-cylinder, that is, a principal Zariski open subset $U$
isomorphic to a product $Z\times\mathbb{A}^{n}$, where $Z$ is a variety. By
[PrZ18], smooth Fano-Mukai fourfolds of genus $10$ are
$\mathbb{A}^{4}$-cylindrical; by [HHT22], a smooth Mukai fourfolds of genus
$8$ is $\mathbb{A}^{1}$-cylindrical; and there exist several families of
codimension at least one of $\mathbb{A}^{1}$-cylindrical Mukai varieties of
genus $g=7,9$ (see [HHT22, PrZ16, PrZ17]). We complete the picture of
cylindricity for Mukai fourfolds of genus $\geq 7$.

######  Theorem 1.2.

Any Mukai fourfold of genus $7,8$ and $9$ is $\mathbb{A}^{2}$-cylindrical.

In [CPW16], it is shown that the affine cone over any plurianticanonically
polarized del Pezzo surface of degree 3 admits no $\mathbb{G}_{a}$-action.
This leads to the notion of generic flexibility of affine varieties111That is,
the special automorphism group acts infinitely transitive on an open orbit of
the variety. which has attracted attention quite recently (see [AKZ19, Per21,
KimPark21]). In Section 5.2, we can conclude that any Mukai fourfold is
generically flexible.

Finally, we refer to [CPPZ21] for an excellent overview and introduction to
cylindrical varieties and applications to flexible varieties.

#### Outline.

In Section 2, we recall results from adjunction theory that we used throughout
the paper. Moreover, we describe birational maps to Mukai fourfolds of genus
$7,8,9,$ and $10$ based on a single blow up of certain Fano fourfolds along
smooth surfaces and an adjunction map on the blow up. The third section is
devoted to the inverses of these maps. Cylinders are discussed in Section 4,
where the proof of Theorem 1.2 (= Corollary 4.4) is given. Section 5 contains
the main result of this paper, Theorem 1.1 (= Theorem 5.9) and its proof. We
close with the problem to generalize Theorem 1.1 to any Mukai fourfold.

#### Notation and conventions.

Throughout the paper, we work over the complex numbers $\mathbb{C}$ and
$X_{2g-2}$ is a Mukai fourfold of index $2$ and genus $g$. Let
$Y\subset\mathbb{P}^{N}$ be a smooth projective variety of dimension $n$ and
let $d$ be its degree. Then $H_{Y}$ is its the hyperplane class and $K_{Y}$ is
its canonical class. The sectional genus of $Y$, denoted $\pi$, is the genus
of the curve $Y\cap H_{1}\cap\ldots\cap H_{n-1}$, where $H_{1},\ldots,H_{n-1}$
are generic hyperplanes in $\mathbb{P}^{N}$. The adjunction formula reads
$2\pi-2=H_{Y}^{n-1}((n-1)H_{Y}+K_{Y})$.

##  2\. Birational maps to Mukai fourfolds and Adjunction theory

###  2.1. Adjunction theory

We refer to [BeS95] for a detailed history of results about adjunction theory.

######  Theorem 2.1 ([SoV87, (0.1) Theorem.]).

Let $\overline{X}$ be a complex projective manifold of dimension $n\geq 2$ and
let $\overline{L}$ be a very ample line bundle over $\overline{X}$. Then
$K_{\overline{X}}+(n-1)\overline{L}$ is spanned by global sections unless
either

  1. $i)$

$(\overline{X},\overline{L})\cong(\mathbb{P}^{n},\mathcal{O}_{\mathbb{P}^{n}}(1))$
or $(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{n}}(2))$;

  2. $ii)$

$(\overline{X},\overline{L})\cong(Q^{n},\mathcal{O}_{Q^{n}}(1))$ where $Q^{n}$
is a smooth quadric in $\mathbb{P}^{n+1}$

  3. $iii)$

$\overline{X}$ is a $\mathbb{P}^{n-1}$-bundle over a smooth curve and the
restriction of $\overline{L}$ to a fibre is
$\mathcal{O}_{\mathbb{P}^{n-1}}(1)$.

Suppose that $K_{\overline{X}}+(n-1)\overline{L}$ is spanned by global
sections. Then We write
$\Phi_{K_{\overline{X}}+(n-1)\overline{L}}=r\circ\varphi$ for the Stein
factorization of $\Phi_{K_{\overline{X}}+(n-1)\overline{L}}$, so
$\varphi:\overline{X}\to X$ is morphism with connected fibers onto a normal
variety $X$ and $r$ is a finite map.

######  Theorem 2.2 ([SoV87, (0.3) Theorem] or [BeS95, section 7.3]).

Let $\overline{X}$ be a complex projective manifold of dimension $n\geq 2$ and
let $\overline{L}$ be a very ample line bundle over $\overline{X}$. Assume
that $K_{\overline{X}}+(n-1)\overline{L}$ is spanned by global sections and
the adjunction map has $n$-dimensional image. Then $X$ is smooth and
$\varphi:\overline{X}\to X$ expresses $\overline{X}$ as the blowing up of a
finite set $s$ on a projective variety $X$. If
$L=\mathcal{O}_{X}(\varphi(\overline{L}))$ then $L$ and $K_{X}+(n-1)L$ are
ample and

$\varphi^{\ast}(K_{X}+(n-1)L)=K_{\overline{X}}+(n-1)\overline{L}.$

###  2.2. Birational maps to Mukai fourfolds

We will describe birational maps to Mukai fourfolds of genus
$g\in\\{7,8,9,10\\}$ based on a single blow up of certain Fano fourfolds along
smooth surfaces and an adjunction map on the blow up. The numerical data for
our constructions are collected in the following table.

$Y$ | $i(Y)$ | $d(Y)$ | $F$ | $d(F)$ | $\pi(F)$ | $g$
---|---|---|---|---|---|---
$\mathbb{P}^{4}$ | $5$ | $1$ | linkage of a Veronese surface in $\mathbb{P}^{4}$ | $8$ | $6$ | $7$
$Q^{4}\subset\mathbb{P}^{5}$ | $4$ | $2$ | linkage of a Veronese surface in $\mathbb{P}^{5}$ | $8$ | $4$ | $8$
$X_{2\cdot 2}\subset\mathbb{P}^{6}$ | 3 | $4$ | Veronese surface in $\mathbb{P}^{5}$ | $4$ | $0$ | $9$
$X_{1^{3}}\subset\mathbb{G}(1,4)$ | $3$ | $5$ | sextic del Pezzo in $\mathbb{P}^{6}$ | $6$ | $1$ | $10$
Table 1. Fano fourfolds $Y$ containing surfaces $F$ which are mapped
birationally to Mukai fourfolds of genus $g$.

Using the Riemann-Roch theorem and Kodaira vanishing theorem, the surfaces
$F\subset Y$ in Table 1 are contained in a unique hypersurface of degree
$i(Y)-2$. Moreover, by linkage of a Veronese surface in $\mathbb{P}^{4}$ or
$\mathbb{P}^{5}$, we have the following Betti tables for the surfaces $F$ as
in Table 1.

$\displaystyle g=7:\quad\begin{tabular}[]{ c | c c c c c c}&&$0$&$1$&$2$&$3$\\\ \hline\cr 0&&$1$&$\cdot$&$\cdot$&$\cdot$\\\ 1&&$\cdot$&$\cdot$&$\cdot$&$\cdot$\\\ 2&&$\cdot$&$1$&$\cdot$&$\cdot$\\\ 3&&$\cdot$&$4$&$5$&$1$\\\ \end{tabular}\quad g=8:\quad\begin{tabular}[]{ c | c c c c c c}&&$0$&$1$&$2$&$3$\\\ \hline\cr 0&&$1$&$\cdot$&$\cdot$&$\cdot$\\\ 1&&$\cdot$&$2$&$\cdot$&$\cdot$\\\ 2&&$\cdot$&$4$&$9$&$4$\\\ \end{tabular}$ $\displaystyle\quad g=9:\quad\begin{tabular}[]{ c | c c c c c c}&&$0$&$1$&$2$&$3$&$4$\\\ \hline\cr 0&&$1$&$1$&$\cdot$&$\cdot$&$\cdot$\\\ 1&&$\cdot$&$6$&$14$&$11$&$3$\\\ \end{tabular}\quad g=10:\quad\begin{tabular}[]{ c | c c c c c c c}&&$0$&$1$&$2$&$3$&$4$\\\ \hline\cr 0&&$1$&$1$&$\cdot$&$\cdot$&$\cdot$\\\ 1&&$\cdot$&$9$&$25$&$25$&$9$\\\ 2&&$\cdot$&$\cdot$&$\cdot$&$1$&$1$\\\ \end{tabular}$

######  Proposition 2.3.

Let $Y$ be a smooth Fano fourfold of index $i:=i(Y)$ and degree $d(Y)$ in
$\mathbb{P}^{g-3}$ as in Table 1 and let $H$ be an ample divisor on $Y$ whose
class generates $\mathrm{Pic}(Y)$. Suppose that $Y$ contains a smooth surface
$F$ as in Table 1. Let $g$ be an integer as in Table 1 depending on $Y$. Then,
the linear system $|(i-1)H-F|$ of hypersurfaces of degree $i-1$ passing
through $F$ defines a birational map

$\Phi_{(i-1)H-F}:Y\dashrightarrow\hat{X}\subset\mathbb{P}^{g+1},$

where $\hat{X}$ is a fourfold of degree $2g-3$ and sectional genus $g$.
Moreover, the fourfold $\hat{X}$ is _isomorphic_ to the blow up of $Y$ along
$F$.

###### Proof.

By Hironaka’s theorem about elimination of indeterminacies, we can choose a
blowup $\sigma:\overline{X}\to Y$ of $F$ on $Y$ with exceptional divisor $E$
such that $\nu=\Phi_{(i-1)H-F}\circ\sigma:\overline{X}\to\hat{X}$ is a
morphism. We will show that the map $\nu$ is an isomorphism onto a smooth Fano
fourfold $\hat{X}$. We set $\overline{\mathcal{L}}=(i-1)\sigma^{\ast}H-E$ and
will show that $\overline{\mathcal{L}}$ is very ample.

Since $F$ is a scheme-theoretical intersection of hypersurface of degree
$i-1$, the linear system $|\overline{\mathcal{L}}|$ is base point free. Hence,
the divisor

$-K_{\overline{X}}=\sigma^{\ast}H+(i-1)\sigma^{\ast}H-E=\sigma^{\ast}H+\overline{\mathcal{L}}$

is ample, that is, $\overline{X}$ is a Fano fourfold of index $i$ whose rank
of the Picard group ${\mathrm{Pic}}(\overline{X})$ is two. Thus, if $\nu$ is
an isomorphism, then the blowup $\overline{X}$ is mapped onto a smooth Fano
fourfold $\hat{X}$ of index $i$, degree $2g-3$, and sectional genus $g$ (see
Table 2).

Let $D$ be the proper transform of the unique hypersurface of $Y$ that passes
through $F$, and whose degree is $i-2$. We write $D\sim(i-2)\sigma^{\ast}H-kE$
for some $k>0$. Since $F$ is smooth, we can compute
$(\varphi^{\ast}H)^{2}\cdot E^{2}$, $(\varphi^{\ast}H)\cdot E^{3}$, and
$E^{4}$, and whence, we have $0\leq\overline{\mathcal{L}}^{3}\cdot D$ as in
Table 2 by [PrZ17, Lemma 2.3]. It follows that $k=1$ and
$\overline{\mathcal{L}}^{3}\cdot D=1$, and $\overline{\mathcal{L}}^{2}\cdot
D^{2}=-1$, $\overline{\mathcal{L}}\cdot D^{3}=1$.

$g$ | $(\varphi^{\ast}H)^{2}\cdot E^{2}$ | $(\varphi^{\ast}H)\cdot E^{3}$ | $E^{4}$ | $\overline{\mathcal{L}}^{3}\cdot D$ | $D^{4}$ | $\overline{\mathcal{L}}^{4}$
---|---|---|---|---|---|---
$7$ | $-8$ | $-42$ | $-149$ | $-29k+30$ | $4$ | $11$
$8$ | $-8$ | $-30$ | $-77$ | $-23k+24$ | $3$ | $13$
$9$ | $-4$ | $-6$ | $-1$ | $-13k+14$ | $3$ | $15$
$10$ | $-6$ | $-12$ | $-15$ | $-15k+16$ | $2$ | $17$
Table 2. Intersection numbers on blowup $Y$ along the surface $F$.

We will show that the pair $(D,\overline{\mathcal{L}}_{D})$ is isomorphic to
$(\mathbb{P}^{3},\mathcal{O}_{\mathbb{P}^{3}}(1))$. Note that
$\overline{\mathcal{L}}-D\sim\sigma^{*}(H)$. From the short exact sequence

$0\to\overline{\mathcal{L}}-D\to\overline{\mathcal{L}}\to\overline{\mathcal{L}}_{D}\to
0$

we deduce that $h^{0}(\overline{\mathcal{L}}_{D})=4$ since
$h^{1}(\sigma^{*}H)=h^{1}(\sigma^{*}H-K_{\overline{X}}+K_{\overline{X}})=0$ by
Kodaira vanishing. Furthermore, since
$\overline{\mathcal{L}}_{D}^{3}=\overline{\mathcal{L}}^{3}\cdot D=1$, the line
bundle $\overline{\mathcal{L}}_{D}$ is ample and by [BeS95, Thm 3.1.2], we
conclude that
$(D,\overline{\mathcal{L}}_{D})\cong(\mathbb{P}^{3},\mathcal{O}_{\mathbb{P}^{3}}(1))$.

Let us now assume that $\overline{\mathcal{L}}$ is not ample. Then, $\nu$ is a
birational morphism which contracts some curve. By the cone theorem and since
$\overline{X}$ is a Fano variety, we know that there exists an extremal ray
$R$ such that $\overline{\mathcal{L}}\cdot R=0$. Since
$\overline{\mathcal{L}}$ is basepoint free, $|\overline{\mathcal{L}}|$ defines
a birational morphism $\nu:\overline{X}\to\hat{X}$ which coincides actually
with the extremal contraction associated to $R$. Furthermore, the rank of the
Picard group $\mathrm{Pic}(\overline{X})$ is two, and whence,
${\mathrm{Pic}}(\hat{X})\cong\mathbb{Z}$ and we can write
$\overline{\mathcal{L}}=\nu^{\ast}L_{\hat{X}}$, where $L_{\hat{X}}$ is the
ample generator of ${\mathrm{Pic}}(\hat{X})\cong\mathbb{Z}$.

The line bundle $\overline{\mathcal{L}}$ yields a supporting linear function
for the extremal ray generated by the curves contained in the fibers of $\nu$.
Since $\overline{\mathcal{L}}^{4}=$ $2g-3$ $>0$, the class $R$ is not nef. We
have $\sigma^{\ast}H\cdot R>0$ because $K_{\overline{X}}\cdot R<0$ and
therefore, $D\cdot R=\overline{\mathcal{L}}\cdot R-\sigma^{\ast}H\cdot R<0$.
We conclude that $D\cdot C<0$ for any curve $C$ in $\overline{X}$ with $[C]\in
R$ so that the divisor $D$ contains all curves on $\overline{X}$ such that
$[C]\in R$.

We denote $E^{\prime}=\\{x\in\overline{X}\mid\nu\text{ is not isomorphism at
}x\\}$ the locus of curves whose numerical classes are in $R$ that is
contained in the divisor $D$. Since $\overline{\mathcal{L}}^{3}D=1$, we get
that $\dim E^{\prime}\leq 2$ and that $\nu$ is small (elementary) contraction
(note that $\dim E^{\prime}=3$ implies $D=E^{\prime}$ and hence,
$\overline{\mathcal{L}}^{3}D=0$, a contradiction).

Applying [Kaw89, Thm 1.1], we have that $E^{\prime}$ is the disjoint union of
irreducible components $E^{\prime}_{i}$ such that
$E^{\prime}_{i}\cong\mathbb{P}^{2}$ and
$\mathcal{N}_{E^{\prime}_{i}/\overline{X}}\cong\mathcal{O}_{\mathbb{P}^{2}}(-1)\oplus\mathcal{O}_{\mathbb{P}^{2}}(-1)$.
Therefore, we have $\ell(R)=\min\\{-K_{\overline{X}}\cdot C\ |\ C\text{
rational, }[C]\in R\\}=1$, and the restriction $\nu|_{D}$ of $\nu$ on $D$ is a
Mori contraction. By [AnM03, Thm 4.1.2], we have $\nu|_{D}$ is the blow up of
a smooth point in a smooth threefold. Thus, $\textrm{rank }\mathrm{Pic}(D)\geq
2$, contradicting
$(D,\overline{\mathcal{L}}_{D})\cong(\mathbb{P}^{3},\mathcal{O}_{\mathbb{P}^{3}}(1))$.
Finally, the line bundle $\overline{\mathcal{L}}$ is ample. Using the Riemann-
Roch theorem and Kodaira vanishing theorem, we obtain the equality
$\dim|\overline{\mathcal{L}}|=g+1$ and $\overline{\mathcal{L}}$ is very ample.
∎

######  Proposition 2.4.

Let $Y$ be a smooth Fano fourfold of index $i:=i(Y)$ and degree $d(Y)$ in
$\mathbb{P}^{g-3}$ as in Table 1 and let $H$ be an ample divisor on $Y$ whose
class generates $\mathrm{Pic}(Y)$. Suppose that $Y$ contains a smooth surface
$F$ as in Table 1. Let $g$ be an integer as in Table 1 depending on $Y$. Then
we have the following statements.

  1. $i)$

The linear system $|(2i-3)H-2F|$ of hypersurfaces of degree $2i-3$ with double
points along the surface $F$ defines a birational map

$\Phi_{(2i-3)H-2F}:Y\dashrightarrow X\subset\mathbb{P}^{g+2},$

where $X$ is a Mukai fourfold of genus $g$ with
${\mathrm{Pic}}(\overline{X})\cong\mathbb{Z}$.

  2. $ii)$

There is a commutative diagram

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{\varphi}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{(2i-3)H-2F}}$$\textstyle{X\subseteq\mathbb{P}^{g+2}}$$\textstyle{x_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $\sigma$ is the blowup of $F$ with exceptional divisor $E$. Let
$\overline{\mathcal{L}}=(i-1)\sigma^{\ast}H-E$ be the very ample divisor
defining the birational morphism to $\hat{X}$ as in Proposition 2.3. The
adjunction map $\Phi_{K_{\overline{X}}+3\overline{\mathcal{L}}}=\varphi$
expresses $\overline{X}$ as the blowup of a point $x_{0}$ on a smooth Mukai
fourfold $X$ of genus $g$ with exceptional divisor $D$.

  3. $iii)$

Let $L$ be the ample generator of ${\mathrm{Pic}}(X)$. Then we have the
following relations

$\displaystyle\varphi^{\ast}L$ $\displaystyle\sim$
$\displaystyle(2i-3)\sigma^{\ast}H-2E,$ $\displaystyle\quad\quad
D\sim(i-2)\sigma^{\ast}H-E,$ $\displaystyle\sigma^{\ast}H$ $\displaystyle\sim$
$\displaystyle\varphi^{\ast}L-2D,$ $\displaystyle\quad\quad
E\sim(i-2)\varphi^{\ast}L-(2i-3)D.$

  4. $iv)$

$\sigma(D)$ is the unique hypersurface of degree $i-2$ containing $F$.

  5. $v)$

$Y\backslash\sigma(D)\cong X\backslash\varphi(E)$.

######  Remark 2.5.

The base locus of $\Phi_{(2i-3)H-2F}$ is the union of the surface $F$ and
$\ell_{g}$ planes, where $\ell_{g}$ is the number of lines through a general
point on a Mukai fourfolds $X_{2g-2}$ (see also Section 3.1).

###### Proof.

To $i)$ and $ii)$: Let $\overline{\mathcal{L}}=(i-1)\sigma^{\ast}H-E$ be the
very ample line bundle on the smooth variety $\overline{X}$. Then by Theorem
2.1, the line bundle $K_{\overline{X}}+3\overline{\mathcal{L}}$ is spanned by
global sections and

$K_{\overline{X}}+3\overline{\mathcal{L}}=-i\sigma^{\ast}H+E+3\cdot((i-1)\sigma^{\ast}H-E)=(2i-3)\sigma^{\ast}H-2E.$

Using the Riemann-Roch theorem and Kodaira vanishing theorem, we obtain that
$\dim|K_{\overline{X}}+3\overline{\mathcal{L}}|=g+2$. Now we write
$\Phi_{K_{\overline{X}}+3\overline{\mathcal{L}}}=r\circ\varphi$ for the Stein
factorization of $\Phi_{K_{\overline{X}}+3\overline{\mathcal{L}}}$, so
$\varphi:\overline{X}\to X$ is a morphism with connected fibers onto a normal
variety $X$ and $r$ is a finite map. This yields the diagram

$\textstyle{\overline{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{K_{\overline{X}}+3\overline{\mathcal{L}}}}$$\scriptstyle{\sigma}$$\scriptstyle{\varphi}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{(2i-3)H-2F}}$$\textstyle{X^{\prime}\subset\mathbb{P}^{g+2},}$

where $\overline{X}\to X^{\prime}\subset\mathbb{P}^{g+2}$ is given by the
linear system $|K_{\overline{X}}+3\overline{\mathcal{L}}|$. We will show that
$r$ is an isomorphism.

Since $h^{0}(K_{\overline{X}}+2\overline{\mathcal{L}})=1>0$, the image of
$\overline{X}$ is of dimension
$\dim\Phi_{K_{\overline{X}}+3\overline{\mathcal{L}}}(\overline{X})=4$. It
follow from Theorem 2.2 that $X$ is smooth and $\varphi:\overline{X}\to X$
expresses $\overline{X}$ as the blowing up of a finite set $s$ on a projective
variety $X$. Since ${\mathrm{rank}}\ {\mathrm{Pic}}(\overline{X})=2$, the set
$s$ is a point $x_{0}$ and so $\varphi$ is the blowing up of the point $x_{0}$
on a smooth projective variety $X$. Furthermore, we can write
$K_{\overline{X}}+3\overline{\mathcal{L}}=\varphi^{\ast}L$, where $L$ is the
ample generator of ${\mathrm{Pic}}(X)\cong\mathbb{Z}$.

As in Theorem 2.2, we define
$\mathcal{L}=\mathcal{O}_{X}(\varphi(\overline{\mathcal{L}}))$, and get that
$\mathcal{L}$ and $K_{X}+3\mathcal{L}$ are ample. Furthermore,
$\overline{\mathcal{L}}=\varphi^{\ast}(\mathcal{L})-\mathcal{O}(\varphi^{-1}(x_{0}))$.
Since $|K_{\overline{X}}+3\overline{\mathcal{L}}|=|(2i-3)\sigma^{\ast}H-2E|$
defines a birational morphism $\overline{X}\to X$, which coincides actually
with the map $\varphi$, we have

$\mathcal{O}(\varphi^{-1}(x_{0}))=\varphi^{\ast}(\mathcal{L})-\overline{\mathcal{L}}=((2i-3)\sigma^{\ast}H-2E)-((i-1)\sigma^{\ast}H-E)=(i-2)\sigma^{\ast}H-E\sim
D.$

Thus, $\varphi$ is birational and its exceptional locus coincides with $D$. In
particular, it is an irreducible divisor.

Since $K_{\overline{X}}=-i\sigma^{\ast}H+E$,
$\overline{\mathcal{L}}=(i-1)\sigma^{\ast}H-E$ and $D=(i-2)\sigma^{\ast}H-E$,
we get that

$K_{\overline{X}}=-2(K_{\overline{X}}+3\overline{\mathcal{L}})+3D,$

and $-K_{X}=2L$. Hence, $-K_{X}$ is an ample Cartier divisor divisible by $2$
in ${\mathrm{Pic}}X$. We conclude that $X$ is a Mukai fourfold. The morphism
$r:X\to X^{\prime}\subset\mathbb{P}^{g+2}$ is given by the linear system
$|L|=|-\frac{1}{2}K_{X}|$. This is an isomorphism by [PrZ17, Lemma 5.6].

In the remaining proof, we identify $X$ with $X^{\prime}$ and
$\varphi_{|3\rho^{\ast}H-2E|}$ with $\varphi$. Since
$K_{\overline{X}}+3\overline{\mathcal{L}}=(2i-3)\sigma^{\ast}H-2E$, we have

$2g(X)-2=(K_{\overline{X}}+3\overline{\mathcal{L}})\cdot\overline{\mathcal{L}}^{3}$

and the sectional genus of $\overline{X}$ is $g$. Thus, $\varphi$ is the
blowup of a point $x_{0}$ on a smooth Mukai fourfold $X$ of genus $g$ with
${\mathrm{Pic}}(X)\cong\mathbb{Z}$ and exceptional divisor $D$. This proves
$i)$ and $ii)$.

To $iii)$ and $iv)$: From the relations $E\sim(i-2)\varphi^{\ast}L-(2i-3)D$
and $D\sim(i-2)\sigma^{\ast}H-E$ in ${\overline{X}}$, we deduce that the
images $\varphi(E)$ and $\sigma(D)$ are hypersurface sections of $X$ and $Y$,
respectively. Furthermore, $\varphi(E)$ is singular along $x_{0}$ and
$\sigma(D)$ is the unique hypersurface section of degree $i-2$ which contains
$F$.

To $v)$: Since $F\subset\sigma(D)$ we have isomorphisms

$X\backslash\varphi(E)\cong{\overline{X}}\backslash(E\cup
D)\cong{Y}\backslash(F\cup\sigma(D))\cong Y\backslash\sigma(D).$

∎

##  3\. Elementary rational maps with center at a point

In the previous section, we described birational maps to Mukai fourfolds of
genus $7,8,9$ and $10$. We will show that the inverses of these maps are given
by tangential projections from a general point of these Mukai fourfolds.
Therefore we describe lines on a Mukai fourfold passing through a fixed point
and then analyze the tangential projection from this point.

###  3.1. Lines through a general point on a Mukai fourfolds

Let $X_{2g-2}$ be a Mukai fourfold of genus $g\in\\{7,8,9,10\\}$ and let $p$
be a general point on $X_{2g-2}$. What is the intersection of $X_{2g-2}$ and
its tangent space $T_{p}(X_{2g-2})$ at $p$? The answer to this question is
well-known and follows from a result of Mukai and Landsberg–Manivel.

In [Muk89], Mukai gave a description of smooth Mukai fourfolds of genus $g=7$,
$8$, $9$, $10$ in terms of linear sections of appropriate rational homogeneous
varieties $M_{g}^{d}$, so-called Mukai models of sectional genus $g$ and
dimension $d$ (see also the treatment in [Kap18]). An anti-canonical embedded
Mukai fourfold $X_{2g-2}$ of genus $g$ is a complete intersection of
$M_{g}^{d}$ and a $\mathbb{P}^{g+2}$. The intersection of $X_{2g-2}$ and its
tangent space $T_{p}(X_{2g-2})$ consists of $\ell_{g}$ lines for a general
point $p$. Indeed, this can be deduced from the fact that the intersection
$\mathbb{T}_{p}\cap M_{g}^{d}$ of the Mukai model $M_{g}^{d}$ and its tangent
space $\mathbb{T}_{p}$ at a point $p$ is a cone over some variety of degree
$\ell_{g}$ which is of codimension $3$ in $\mathbb{T}_{p}$. Hence, the
intersection $\mathbb{P}^{g+2}\cap\mathbb{T}_{p}\cap M_{g}^{d}$ is one-
dimensional and consist exactly of $\ell_{g}$ lines. In the following table,
we may summarize the results of this section which follow from [LM03, Theorem
4.8].

$g$ | $M_{g}^{d}$ | $M_{g}^{d}\cap\mathbb{T}_{p}$ | $\ell_{g}$
---|---|---|---
$7$ | $M_{7}^{10}=\mathrm{OG}(5,10)$ | cone over $\mathbb{G}(1,4)\subset\mathbb{P}^{9}$ | 5
$8$ | $M_{8}^{8}=\mathbb{G}(1,5)$ | cone over $\mathbb{P}^{1}\times\mathbb{P}^{3}$ | 4
$9$ | $M_{9}^{6}=\mathrm{LG}(3,6)$ | cone over the Veronese surface | 4
$10$ | $M_{10}^{5}=\mathrm{G}_{2}$ | cone over twisted cubic curve | 3
Table 3.  Mukai fourfolds of genus $g=7$, $8$, $9$, $10$.

###  3.2. Terminology and tangential projections

Let $X:=X_{2g-2}\subseteq\mathbb{P}^{g+2}$ be a Mukai fourfold of genus
$g\in\\{7$, $8$, $9$, $10\\}$ with ${\mathrm{Pic}}(X)\cong\mathbb{Z}L$, and
let $L$ be the hyperplane class of $X$. Let $x_{0}\in X$ be a general point on
$X$, and let $\mathcal{L}_{x_{0}}$ be the union of lines passing through
$x_{0}$. We denote $\ell_{g}$ the number of lines through a point. Then, we
have two birational models of the blow up of $X$ in $x_{0}$ as follows:

  1. $\bullet$

Let $\varphi^{\prime}:X^{\prime}\to X$ be a blowup of $x_{0}$ on $X$ with
exceptional divisor $D^{\prime}$. We use $L^{\prime}$ to denote the pullback
of the hyperplane class $L$ to $X^{\prime}$. Let
$\mathcal{L}^{\prime}_{x_{0}}$ be the proper transform of
$\mathcal{L}_{x_{0}}$. Then, $\mathcal{L}^{\prime}_{x_{0}}$ is the disjoint
union of $\ell_{g}$ lines $\\{\ell_{i}^{\prime}\\}_{i=1}^{\ell_{g}}$ on
$X^{\prime}$ and each $\ell_{i}^{\prime}$ cut $D^{\prime}$ along a point
$x_{i}$.

  2. $\bullet$

We blow up $X^{\prime}$ along $\mathcal{L}^{\prime}_{x_{0}}$ to obtain
$\widetilde{X}$ and introduce ${\ell_{g}}$ exceptional divisors
$D_{1},\ldots,D_{\ell_{g}}$ isomorphic to $\mathbb{P}^{3}$. Let
$\widetilde{D}$ be the proper transforms of $D^{\prime}$ and let
$\widetilde{L}$ be the pullback of $L^{\prime}$ to $\widetilde{X}$. Then
$\widetilde{D}$ is isomorphic to $\mathbb{P}^{3}$ blown up along $\ell_{g}$
points $\\{x_{i}\\}_{i=1}^{\ell_{g}}\subset D^{\prime}$ with exceptional
divisors $\\{\widetilde{P_{i}}\\}_{i=1}^{\ell_{g}}$. Thus, each $D_{i}$ cut
$\widetilde{D}$ along a plane $\widetilde{P_{i}}\cong\mathbb{P}^{2}$.

We have the following intersection numbers.

######  Lemma 3.1.

We have $\widetilde{L}^{4}=2g-2$,
$\widetilde{L}\widetilde{D}=\widetilde{L}D_{i}=0$, $D_{i}^{4}=1$,
$\widetilde{D}^{4}={\ell_{g}}-1$,
$\widetilde{D}^{3}D_{i}=\widetilde{D}D_{i}^{3}=-1$,
$\widetilde{D}^{2}D_{i}^{2}=1$ for all $i=1,\ldots,4$.

###### Proof.

First, $\widetilde{L}D=\widetilde{L}D_{i}=0$ since their intersection is
empty. It’s clear that $\widetilde{L}^{4}=2g-2$.
Notice that $\widetilde{D}$ is isomorphic to $D^{\prime}\cong\mathbb{P}^{3}$
blown up along the points $\\{x_{i}\\}_{i=1}^{\ell_{g}}$ with the exceptional
divisors $\\{\widetilde{P_{i}}\\}_{i=1}^{\ell_{g}}$. Thus each $D_{i}$ cut
$\widetilde{D}$ along a plane $\widetilde{P_{i}}\cong\mathbb{P}^{2}$. Write
$\mathrm{Pic}(\widetilde{D})=\langle\widetilde{H},\widetilde{P_{1}},\ldots,\widetilde{P_{\ell_{g}}}\rangle$
where $\widetilde{H}$ is the polarization from $\mathbb{P}^{3}$ while
$\widetilde{P_{i}}$ are the exceptional divisors over the points $x_{i}$.
Since $N_{x_{i}/\mathbb{P}^{3}}=\mathcal{O}_{x_{i}}^{3}(1)$ then write
$\xi=c_{1}(\mathcal{O}_{\mathbb{P}(N_{x_{i}/\mathbb{P}^{3}})}(-1))$ and
equality $\xi=\widetilde{P_{i}}|_{\widetilde{P}_{i}}$ we get that $\xi^{2}=1$
in the Chow group of $\widetilde{P_{i}}=\mathbb{P}(N_{x_{i}/\mathbb{P}^{3}})$.
We have $N_{\widetilde{D}/\widetilde{X}}=\mathcal{O}(\widetilde{H})$ and
$D_{i}|_{\widetilde{D}}=-\widetilde{P}_{i}$. Thus we obtain

$\displaystyle D_{i}^{3}\widetilde{D}$
$\displaystyle=(-\widetilde{P}_{i})^{3}=-1,$ $\displaystyle
D_{i}^{2}\widetilde{D}^{2}$ $\displaystyle=(-\widetilde{P}_{i})^{2}H=1,$ and
$\displaystyle D_{i}\widetilde{D}^{3}$
$\displaystyle=(-\widetilde{P}_{i})H^{2}=-1.$

Now, since $\varphi^{\prime}:X^{\prime}\to X$ is a blowup of $x_{0}$ on $X$
with exceptional divisor $D^{\prime}$, we have
$\rho^{\ast}(D^{\prime})=\widetilde{D}+\sum\limits_{i=1}^{\ell_{g}}D_{i}$ and
so that $\widetilde{D}^{4}={\ell_{g}}-1$ because $D^{\prime 4}=(-1)^{4-1}$. It
follows from $K_{X^{\prime}}=\varphi^{\prime\ast}K_{X}-3D^{\prime}$ and
[Ful98, section 15.4] that $D_{i}^{4}=1$. ∎

We now consider the morphism induced by the linear system
$|\widetilde{L}-\sum_{i=1}^{\ell_{g}}D_{i}|$. By Lemma 3.1, this morphism
contracts each $D_{i}$ to a $\mathbb{P}^{2}$. Geometrically, each $D_{i}$ is a
$\mathbb{P}^{1}$-bundle over $\mathbb{P}^{2}$. Indeed, fix a general point
$p_{i}\in\widetilde{P_{i}}$ and through each $p_{i}\neq p^{\prime}_{i}\in
D_{i}$ passes a unique line $l$ intersecting $P_{i}$. The bundle map is given
by $p^{\prime}_{i}\mapsto l\cap\widetilde{P_{i}}$. The blow down of each
$D_{i}$ to $\mathbb{P}^{2}$ is given by the above linear system. The resulting
$\overline{X}$ is birational to $X^{\prime}=\textrm{Bl}_{x_{0}}(X)$. Then we
organize these maps into a diagram:

(7)

where $\varphi:\overline{X}\to X$ is the another blow up of $X$ along $x_{0}$
with the exceptional divisor $D$. Note that each plane $P_{i}$ for
$i=1,\ldots,{\ell_{g}}$ is contained in $D$ and therefore, contracted under
$\varphi$.

In the following, we consider a tangential projection of a Mukai fourfold from
a general point $x_{0}$ on it such that there are exactly $\ell_{g}$ lines
through $x_{0}$ as in Section 3.1. On the blow up $\widetilde{X}$, the
tangential projection is given by the linear system

$|\widetilde{L}-\sum_{i=1}^{\ell_{g}}D_{i}-2\widetilde{D}|.$

######  Proposition 3.2.

Let $X\subseteq\mathbb{P}^{g+2}$ be a Mukai fourfold of genus $g=7$, $8$, $9$,
$10$ with ${\mathrm{Pic}}(X)\cong\mathbb{Z}L$, and let $L$ be the hyperplane
class of $X$. Let $x_{0}\in X$ be a general point, and let
$\mathcal{L}_{x_{0}}$ be the union of lines passing through $x_{0}$. Then we
have the following statements.

  1. $i)$

The tangential projection from the point $x_{0}$ defines a birational map

$\Phi_{x_{0}}:X\dashrightarrow Y,$

where $Y$ is a smooth Fano variety of index $i:=i(Y)$ and degree $d(Y)$ in
$\mathbb{P}^{g-3}$ as in Table 1. Let $\overline{X}$ be in (7), that is, the
blow up of $X$ in $x_{0}$ with exceptional divisor $D$.  There is a
commutative diagram

$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\sigma}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{x_{0}}}$$\textstyle{Y}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $\sigma$ is the blowup of a smooth surface $F$ as in Table 1 with
exceptional divisor $E$.

  2. $ii)$

Let $H$ be the ample generator of ${\mathrm{Pic}}(Y)$. Then we have the
following relations

$\displaystyle\varphi^{\ast}L$ $\displaystyle\sim$
$\displaystyle(2i-3)\sigma^{\ast}H-2E,$ $\displaystyle\quad\quad
D\sim(i-2)\sigma^{\ast}H-E,$ $\displaystyle\sigma^{\ast}H$ $\displaystyle\sim$
$\displaystyle\varphi^{\ast}L-2D,$ $\displaystyle\quad\quad
E\sim(i-2)\varphi^{\ast}L-(2i-3)D.$

  3. $iii)$

$X\backslash\varphi(E)\cong Y\backslash\sigma(D)$.

  4. $iv)$

$\sigma(D)$ is a singular hypersurface section of $Y$ with its degree $i-2$
and its singular locus coincides with the locus of points $p\in\sigma(D)$,
such that the restriction $\sigma|_{D}:D\to\sigma(D)$ is not an isomorphism
over $p$.

  5. $v)$

Let $g=7$. Then $\varphi(E)$ coincides with the union of conic curves in $X$
passing through $x_{0}$ and for all conic curves $C$ passing through $x_{0}$,
$C\subset\varphi(E)$.

In particular, the rational map $\Phi_{x_{0}}$ is the inverse of the rational
map $\Phi_{(2i-3)H-2F}:Y\dashrightarrow X\subset\mathbb{P}^{g+2}$ as in
Proposition 2.4.

###### Proof.

To $i)$ and $ii)$: Recall that
$\widetilde{L}-\sum\limits_{i=1}^{\ell_{g}}D_{i}$ on $\widetilde{X}$ defines a
birational morphism $\widetilde{X}\to\overline{X}$ which coincides with the
map $\rho$. It is clear that

$-K_{\overline{X}}=-(\varphi^{\prime}\circ\rho^{\prime})^{\ast}K_{X}+3D+2\sum\limits_{i=1}^{4}D_{i}=-2(L-\sum\limits_{i=1}^{4}D_{i})+3D.$

Thus we have $K_{\overline{X}}=-2\varphi^{\ast}L$, that is, $\overline{X}$ is
a Fano fourfold whose rank of ${\mathrm{Pic}}(\overline{X})$ is two. By the
Cone theorem, there exists a Mori contraction $\sigma:\overline{X}\to U$
different from $\varphi$. We organize these maps into a diagram:

$\textstyle{\widetilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{\prime}}$$\scriptstyle{\rho}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{\prime}}$$\textstyle{\overline{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\sigma}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{x_{0}}}$$\textstyle{U}$

By Lemma 3.1, we have $(\varphi^{\ast}L)^{4}=2g-2+\ell_{g}$, and
$(\varphi^{\ast}L)^{3}\cdot D=\ell_{g}$, $(\varphi^{\ast}L)^{2}\cdot
D^{2}=\ell_{g}$, $(\varphi^{\ast}L)\cdot D^{3}=\ell_{g}$ and
$D^{4}=\ell_{g}-1$. Therefore, we have
$(\varphi^{\ast}L-2D)^{3}\cdot((i-2)\varphi^{\ast}L-(2i-3)D)=0$ and
$(\varphi^{\ast}L-2D)^{4}=d(Y)$. This means that the divisor class of
$\varphi^{\ast}L-2D$ is not ample, and so it yields a supporting linear
function of the extremal ray generated by the curves in the fibers of
$\varphi$. Moreover, we can write $\varphi^{\ast}L-2D=\sigma^{\ast}H$, where
$H$ is the ample generator of ${\mathrm{Pic}}(U)\cong\mathbb{Z}$. By the
Riemann-Roch theorem and Kodaira vanishing theorem, we have
$\dim|\varphi^{\ast}L-2D|=4$. Thus, $\varphi^{\ast}L-2D$ defines a birational
morphism $\overline{X}\to Y$ which coincides with the map $\sigma$. The
birationality of $\Phi_{x_{0}}$ follows. This prove $i)$.

Since $\dim|\varphi^{\ast}L-3D|=0$ and
$(\varphi^{\ast}L-2D)^{3}\cdot((i-2)\varphi^{\ast}L-(2i-3)D)=0$, the linear
system $|(i-2)\varphi^{\ast}L-(2i-3)D)|$ contains a unique divisor $E$
contracted by $\sigma$. Since rank of ${\mathrm{Pic}}(\overline{X})$ is two,
the divisor $E$ is irreducible and the relations in $iii)$ follow.

Since $(\varphi^{\ast}L-2D)^{2}\cdot E^{2}=-d(F)$, the image $F=\sigma(E)$ is
a surface with $\deg F=H^{2}\cdot\sigma(E)=d(F)$ (see Table 1). Furthermore,
the sectional genus of $F$ as given in Table 1 can be computed using [PrZ17,
Lemma 2.3] or [IsP99, Lemma 2.2.14].

Since ${\mathrm{rank}}\ {\mathrm{Pic}}(\overline{X})=2$, the exceptional locus
of $\sigma$ coincides with $E$, and $E$ is a prime divisor. Therefore,
$\sigma$ has at most a finite number of $2$-dimensional fibers. By the main
theorem in [AnW98], $F$ has at most isolated singularities. Furthermore, since
$E^{4}=(\varphi^{\ast}L-3D)^{4}=-1$ and the classification of surfaces of
degree $\geq 8$ implies that $F$ is a smooth surface as in Table 1 contained
in $\sigma(D)$. Hence the morphism $\sigma:\overline{X}\to Y$ is the blowup of
the smooth surface $F$, as stated in $iii)$.

To $iii)$: Since $F\subset\sigma(D)$ we have isomorphisms

$X\backslash\varphi(E)\cong{\overline{X}}\backslash(E\cup
D)\cong{Y}\backslash(F\cup\sigma(D))\cong Y\backslash\sigma(D).$

To $iv)$: From the relations $E\sim(i-2)\varphi^{\ast}L-(2i-3)D$ and
$D\sim(i-2)\sigma^{\ast}H-E$ in $\overline{X}$, we deduce that the images
$\varphi(E)$ and $\rho(D)$ are a hyperplane sections of $X$ and $Y$,
respectively. The hypersurface $\sigma(D)$ of degree $i-2$ is singular because
it contains the surface $F$. Since $-K_{D}\sim
2(\varphi^{\ast}L-2D)|_{D}\sim\sigma^{*}(2H)|_{D}$, the restriction
$\rho|_{D}:D\to\rho(D)$ is a crepant morphism.

To $v)$: The linear projection $\Phi_{x_{0}}:X\dashrightarrow Y$ sends
$\varphi(E)$ to a smooth surface $F$ in Y contracting the conic curve passing
through $x_{0}$ in $\varphi(E)$. These conics correspond to the rulings of
$\sigma|_{E}:E\to F$. It follows that $\varphi(E)$ is covered by such conics.
Since through any point in $\varphi(E)$ passes a unique ruling of
$\sigma|_{E}:E\to F$, given a point $x_{0}\neq p\in\varphi(E)$ there is a
unique conic in $\varphi(E)$ through $p$ and $x_{0}$.

Any curve of degree $d$ through the point $x_{0}$ will be either tangent to
the tangent space at $q_{0}$ or will be contained in it (as the five lines).
In the first case, this means that the intersection multiplicity is at least
$2$ and the projected curve has degree at most $d-2$. Hence, the image of a
conic curve passing through $x_{0}$ has degree $0$ which means it is
contracted to a point. For the second case, since $x_{0}$ is a general point,
intersection of the tangent space of $X$ at $x_{0}$ with X is the five lines.
Thus a conic curve passing through $x_{0}$ belong in $\varphi(E)$.

∎

##  4\. $\mathbb{A}^{2}$-cylinders on Mukai fourfolds of $g=7$, $8$, $9$

In this section, we construct an open subset of any Mukai fourfold $X_{2g-2}$
of genus $g\in\\{7,8,9\\}$ that can be covered by $\mathbb{A}^{2}$-cylinders.
Let us recall the definition of $\mathbb{A}^{n}$-cylinder.

######  Definition 4.1.

An $\mathbb{A}^{n}$-cylinder in $X$ is a pair $(Z,\varphi)$ and an open
embedding $\varphi:Z\times\mathbb{A}^{n}\to X$ where $Z$ is a variety and
$\mathbb{A}^{n}$ is the affine $n$-space over $\mathbb{C}$. We say that $X$ is
$\mathbb{A}^{n}$-cylindrical if there exists an $\mathbb{A}^{n}$-cylinder
$(Z,\varphi)$ in $X$. Given a divisor $H\subset X$ we say that a
$\mathbb{A}^{n}$-cylindrical $(Z,\varphi)$ in $X$ is _$H$ -polar_ if
$\varphi(Z\times\mathbb{A}^{n})=X\backslash\textrm{supp}D$ for some effective
divisor $D\in|kH|$ where $k>0$. ([KPZ11, Definitions 3.1.5, 3.1.7])

In the first step, we show that for a Mukai fourfold of genus
$g\in\\{7,8,9\\}$, there exists a hypersurface $Y\subset X_{2g-2}$ and a
singular cubic $W$ such that $X_{2g-2}\backslash
Y\cong\mathbb{P}^{4}\backslash W$. The second step is to construct an explicit
$\mathbb{A}^{2}$-cylinder covering of the complement of a nodal cubic
threefold in $\mathbb{P}^{4}$. The proof follows ideas of [HHT22, Proposition
5.2].

######  Proposition 4.2.

Let $X_{2g-2}$ be a Mukai fourfold for $g\in\\{7,8,9\\}$. There exists an open
subset $U$ of $X$ such that

  1. $i)$

$U=X_{2g-2}\backslash Y$, where $Y$ is a cubic hypersurface section of
$X_{2g-2}$.

  2. $ii)$

There exists a singular cubic threefold $W\subset\mathbb{P}^{4}$, and a
birational map $\Phi:X_{2g-2}\dashrightarrow\mathbb{P}^{4}$ such that

$X_{2g-2}\backslash Y\cong\mathbb{P}^{4}\backslash W.$

###### Proof.

Case $g=7$: Proposition 3.2. Note that the hypersurface $Y$ is the unique
cubic hypersurface that intersects $X_{12}$ in a point with multiplicity at
least $7$. The cubic threefold $W$ is the unique cubic threefold with the
maximal number of 10 nodes.

Case $g=8$: This case follows immediately from [HHT22, Proposition 3.1] and
the cubic threefold $W$ is again the unique cubic threefold with the maximal
number of 10 nodes. Note that the base locus of the birational map from
$\mathbb{P}^{4}$ to $X_{14}$ is a surface of degree $7$ and sectional genus
$4$.

Case $g=9$: This case follows from Proposition 3.2 and [PrZ16, Theorem 3.1].
Indeed, by Proposition 3.2 we find a hyperplane $H\subset\mathbb{P}^{11}$ and
a hyperplane $H^{\prime}\in\mathbb{P}^{6}$ such that $X_{16}\backslash H\cong
X_{2\cdot 2}\backslash H^{\prime}$. In [PrZ16, Theorem 3.1] the author show
that the complement of a quadric hypersurface section of a complete
intersection of two quadrics $X_{2\cdot 2}\subset\mathbb{P}^{6}$ is isomorphic
to the complement of a quadric hypersurface in $\mathbb{P}^{4}$. Therefore,
the hypersurfaces $Y$ and $W$ as stated in the proposition are the union of a
hyperplane and a quadric hypersurface. ∎

######  Proposition 4.3.

Let $W$ be a singular cubic hypersurface in $\mathbb{P}^{4}$ and let $p$ be an
ordinary double point of $W$. Then for a point $x\in\mathbb{P}^{4}\backslash
W$ there exists a hyperplane $P$, a quadric $Q$ and a principal affine open
subset $U=U_{x}\subset P^{4}\backslash W$ such that

  1. $i)$

$x\in U$;

  2. $ii)$

$U\cong\mathbb{A}^{2}\times Z$, where $Z$ is an affine surface.

  3. $iii)$

$U=\mathbb{P}^{4}\backslash(P\cup Q\cup W)$.

###### Proof.

Step 1: Let $W=V(f)$ be generated by a cubic equation $f$, and let
$\mathcal{I}_{p}(1)=(l_{1},l_{2},l_{3},l_{4})$ be the linear forms generating
the ideal of $p$. We choose a rank $4$ quadric $Q=V(g)$ that is singular in
$p$ (that is, $g\in|\mathcal{I}^{2}_{p}(2)|$). Then, we consider the following
cubic Cremona transformation

$\textstyle{\Psi_{Q}:\mathbb{P}^{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{|g\cdot
l_{1},g\cdot l_{2},g\cdot l_{3},g\cdot l_{4},f|}$$\textstyle{\mathbb{P}^{4}.}$

The image of $W$ is a hyperplane $P_{W}$ of $\mathbb{P}^{4}$, and the image of
$Q$ is point which does not lie on $P_{W}$. The total transform of the point
$p$ is another rank $4$ quadric $Q^{\prime}$ whose vertex is the image of $Q$.
Note that the inverse of $\Psi_{Q}$ is also a cubic Cremona transformation
given in a similar form as $\Psi_{Q}$ such that the total transform of the
base locus of the inverse of $\Psi_{Q}$ is the quadric $Q$. We get

$\mathbb{P}^{4}\backslash(W\cup Q)\cong\mathbb{P}^{4}\backslash(P_{W}\cup
Q^{\prime}).$

Varying the quadric hypersurface $Q$, we can cover $\mathbb{P}^{4}\backslash
W$ by open sets of the form $\mathbb{P}^{4}\backslash(W\cup Q)$.

Step 2: Now, we will cover the open set $\mathbb{P}^{4}\backslash(W\cup Q)$ by
open $\mathbb{A}^{2}$-cylinders. Therefore, we perform a second quadric
Cremona transformation. Let $q\in P_{W}\cap Q^{\prime}$ be an intersection
point and let
$\mathcal{I}_{q}(1)=(l_{1}^{\prime},l_{2}^{\prime},l_{3}^{\prime},l_{4}^{\prime})$
be the linear forms generating its ideal. Let $Q^{\prime}=V(g^{\prime})$ be
generated by a quadratic equation $g^{\prime}$. Let $P^{\prime}=V(h^{\prime})$
be the tangent hyperplane to $Q^{\prime}$ at the point $q$. Then, we consider
the following quadric Cremona transformation

$\Psi_{Q,q}:\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
7.80278pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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The image of $Q^{\prime}$ is a hyperplane $P_{1}\subset\mathbb{P}^{4}$.
Furthermore, the total transform of the base locus of $\Psi_{Q,q}$ is another
hyperplane $P_{2}$. The image of $P_{W}$ is a third hyperplane $P_{3}$. Note
that the inverse of $\Psi_{Q,q}$ is again a quadric Cremona transformation
whose total transform of its base locus is the tangent hyperplane
$P^{\prime}$. We get

$\mathbb{P}^{4}\backslash(P^{\prime}\cup P_{W}\cup
Q^{\prime})\cong\mathbb{P}^{4}\backslash(P_{1}\cup P_{2}\cup P_{3})$

Let $\ell=P_{1}\cap P_{2}\cap P_{3}$ be the intersection.

Then the projection $\pi_{\ell}:\mathbb{P}^{4}\dashrightarrow\mathbb{P}^{2}$
with center $\ell$ defines an $\mathbb{A}^{2}$-cylinder on
$\mathbb{P}^{4}\backslash(P_{1}\cup P_{2}\cup P_{3})$. Varying again the point
$q$ and therefore, the tangent hyperplane $P^{\prime}$, we can cover
$\mathbb{P}^{4}\backslash(P_{W}\cup Q^{\prime})$ by
$\mathbb{A}^{2}$-cylinders.

Therefore, $\mathbb{P}^{4}\backslash W$ can be covered by
$\mathbb{A}^{2}$-cylinders of the form $\mathbb{P}^{4}\backslash(P\cup Q\cup
W)$ where $P$ is the preimage of the hyperplane $P^{\prime}$ as in Step 2.
Note that $P$ is again a hyperplane since $P^{\prime}$ contains the vertex of
$Q^{\prime}$. ∎

######  Corollary 4.4.

Any Mukai fourfold of genus $7,8$ and $9$ is $\mathbb{A}^{2}$-cylindrical.

##  5\. On the flexibility of Mukai fourfolds of $g=7$, $8$, $9$

We recall the definition of a transversal cover of cylinders and construct a
transversal $\mathbb{A}^{2}$-cylinder covering of the complement
$X_{2g-2}\backslash Y$ of a Mukai fourfold of genus $g$ and a hypersurface
$Y$. Using the computer algebra system Macaulay2 (see [macaulay2]), we
conclude that the general Mukai fourfold is flexible.

######  Definition 5.1.

  1. $i)$

([Per13, Definitions 3]) A subset $Y\subset X$ is called _invariant_ with
respect to a $\mathbb{A}^{n}$-cylinder $(Z,\varphi)$ in $X$ if

$Y\cap\varphi(Z\times\mathbb{A}^{n})=\varphi(\pi_{1}^{-1}(\pi_{1}(\varphi^{-1}(Y)))),$

where $\pi_{1}:Z\times\mathbb{A}^{n}\to Z$ is the first projection of the
direct product.

  2. $ii)$

([Per13, Definitions 4]) We say that a variety $X$ is transversally covered by
$\mathbb{A}^{n}$-cylinders ${\\{(Z_{i},\varphi_{i})\\}_{i\in I}}$ in $X$ if
$X$ has a covering

(8) $X=\bigcup\limits_{i\in I}U_{i}$

where each $U_{i}$ is a Zariski open subset in $X$ such that
$U_{i}=\varphi_{i}(Z_{i}\times\mathbb{A}^{n})$ and it does not admit any
proper invariant subset with respect to every $\mathbb{A}^{n}$-cylinders
$(Z_{i},\varphi_{i})$. A subset $Y\subset X$ is proper if it is nonempty and
different from $X$.

###  5.1. Transversality of the $\mathbb{A}^{2}$-cylinder covering

We choose hypersurfaces $Y$ such that the Mukai fourfold $X_{2g-2}$ and a
collection of hyperplanes $P$ and quadric hypersurfaces $Q$ as in Lemma 4.3
such that $X_{2g-2}\backslash
Y\stackrel{{\scriptstyle\Phi}}{{\cong}}\mathbb{P}^{4}\backslash W$ is covered
by open $\mathbb{A}^{2}$-cylinders $V_{Q,P}:=\Phi^{-1}(U_{Q,P})$ where

$U_{Q,P}:=\mathbb{P}^{4}\backslash(P\cup Q\cup
W)\cong\mathbb{P}^{4}\backslash(P^{1}_{Q,P}\cup P^{2}_{Q,P}\cup P^{3}_{Q,P}).$

We recall some notation from the proof of Lemma 4.3 that we need in the proof
of the transversality of the covering $\\{V_{Q,P}\\}_{P,Q}$ of
$X_{2g-2}\backslash Y$.

For the open subset $V_{Q,P}$, let $W$ be the singular cubic hypersurface in
$\mathbb{P}^{4}$, and let $Q$ be a rank $4$ quadric hypersurface of
$\mathbb{P}^{4}$ with vertex in a node of $W$. There exists a hyperplane
$P_{W}$ and a quadric $Q^{\prime}$ together with a cubic Cremona
transformation, denoted by
$\Psi_{Q}:\mathbb{P}^{4}\dashrightarrow\mathbb{P}^{4}$ such that

$\mathbb{P}^{4}\backslash(W\cup Q)\cong\mathbb{P}^{4}\backslash(P_{W}\cup
Q^{\prime}).$

For the hyperplane $P$, let $P^{\prime}=\Psi_{Q}(P)$ be its image, there exist
three hyperplanes $\\{P^{j}_{Q,P}\\}_{j=1}^{3}$ and a quadric Cremona
transformation, denoted by
$\Psi_{Q,P}:\mathbb{P}^{4}\dashrightarrow\mathbb{P}^{4}$ such that

$\mathbb{P}^{4}\backslash(P_{W}\cup Q^{\prime}\cup
P^{\prime})\cong\mathbb{P}^{4}\backslash(P^{1}_{Q,P}\cup P^{2}_{Q,P}\cup
P^{3}_{Q,P})$

as in Lemma 4.3. Let $\ell_{Q,P}=P^{1}_{Q,P}\cap P^{2}_{Q,P}\cap P^{3}_{Q,P}$
be the intersection. Then the structure of an $\mathbb{A}^{2}$-cylinder on
$V^{i}_{Q,P}$ is induced by the projection
$\pi_{Q,P}:\mathbb{P}^{4}\dashrightarrow\mathbb{P}^{2}$ with center
$\ell_{Q,P}$. In summary, we have the following chain of birational maps

$\textstyle{X_{2g-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{\mathbb{P}^{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{Q}}$$\textstyle{\mathbb{P}^{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{Q,P}}$$\textstyle{\mathbb{P}^{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{Q,P}}$$\textstyle{\mathbb{P}^{2}.}$

######  Proposition 5.2.

The $\mathbb{A}^{2}$-cylinder covering of $X_{2g-2}\backslash Y$ as above is
transversal.

###### Proof.

Suppose that it is not transversal. Let $v\in V$ be a point on a proper subset
of $V\subset X_{2g-2}$ that is invariant with respect to our covering. For a
cylinder $V_{Q,P}$ containing the point $v$, we replace $V$ by $V\cap V_{Q,P}$
and assume that $V$ is contained in the chosen cylinder. Using the above
notation, we consider the $\mathbb{A}^{2}$-fiber of the cylinder $V_{Q,P}$
containing the image of the point $v$, that is,

$S_{0}:=(\pi_{Q,P})^{-1}((\pi_{Q,P}\circ\Psi_{Q,P}\circ\Psi_{Q}\circ\Phi)(v))$

Let $S$ be the closure of the preimage $\Psi_{Q,P}^{-1}(S_{0})$ which is a
quadric surface since $\Psi_{Q,P}$ is a quadric Cremona transformation. Note
that $S$ is contained in $(\Psi_{Q}\circ\Phi)(V)$ since $V$ is invariant with
respect to $V_{Q,P}$.

######  Claim 5.3.

There exist another cylinder $V_{Q,P_{1}}$ such that the closure of the image

$C:=(\pi_{Q,P_{1}}\circ\Psi_{Q,P_{1}})(S)$

is a line in $\mathbb{P}^{2}$.

###### Proof of the claim..

Note that the intersection of the surface $S$ with the hyperplane $P_{W}$ is
exactly the base locus of $\Psi_{Q,P}$ since $S_{0}$ contains the projection
center $\ell_{Q,P}$ of $\pi_{Q,P}$.

But the surface $S$ does not contain the base locus of $\Psi_{Q,P_{1}}$ for
another cylinder $V_{Q,P_{1}}$ and therefore, the image of $S$ under
$\Psi_{Q,P_{1}}$ is again a quadric surface that does not contain the
projection center $\ell_{Q,P_{1}}$ of $\pi_{Q,P_{1}}$. Hence,
$(\pi_{Q,P_{1}}\circ\Psi_{Q,P_{1}})(S)$ coincides with the image of the linear
span $\langle\Psi_{Q,P_{1}}(S)\rangle\cong\mathbb{P}^{3}$ which is a line as
claimed. ∎

Therefore, the closure of the preimage

$S_{1}:=(\pi_{Q,P_{1}}\circ\Psi_{Q,P_{1}}\circ\Psi_{Q})^{-1}(C)$

is a sextic hypersurface in $\mathbb{P}^{4}$ since
$\Psi_{Q,P_{1}}\circ\Psi_{Q}$ is the composition of a cubic and a quadric
Cremona transformation and is again contained in $\Phi_{i}(V)$ by the
invariance of $V$.

######  Claim 5.4.

There exist another cylinder $V_{Q_{2},P_{2}}$ such that

$S_{2}:=(\pi_{Q_{2},P_{2}}\circ\Psi_{Q_{2},P_{2}}\circ\Psi_{Q_{2}})(S_{1})$

is dense in $\mathbb{P}^{2}$.

###### Proof of the claim..

Notice that the intersection of the hypersurface $S_{1}$ and the quadric $Q$
are six planes passing through the node of $Q$. Furthermore the hypersurface
$\Psi_{Q}(S_{2})$ contains the base locus of $\Psi_{Q,P_{1}}$. But for general
choice of $Q_{2}$ and $P_{2}$, the hypersurface $S_{1}$ and its image do not
contain the base locus of $\Psi_{Q_{2}}$ and $\Psi_{Q_{2},P_{2}}$,
respectively. Hence, the image of $S_{1}$ under the composition
$\Psi_{Q_{2},P_{2}}\circ\Psi_{Q_{2}}$ for another cylinder
$V^{i}_{Q_{2},P_{2}}$ is a hypersurface of degree $24$. Hence, the claim
follows. ∎

Therefore, the preimage
$(\pi_{Q_{2},P_{2}}\circ\Psi_{Q_{2},P_{2}}\circ\Psi_{Q_{2}})^{-1}(S_{2})$ is
dense in $X_{2g-2}$ and is contained the $\Phi(V)$. Since we assumed that $V$
is a proper subset, this is a contradiction. ∎

###  5.2. Generic flexibility of Mukai fourfolds of genus $7,8$ and $9$

######  Definition 5.5 ([AFKKZ13]).

Let $V$ be an affine algebraic variety. A point $p\in V$ is called flexible if
the tangent space $T_{p}V$ is spanned by the tangent vectors to the orbits of
actions of the additive group of the field $\mathbb{G}_{a}$ on $V$ . The
variety $V$ is called flexible if each smooth point of $V$ is flexible . We
also say that $V$ is generically flexible if every point in a non-empty
Zariski open subset of $V$ is flexible.

Now, we recall the flexibility criterion for affine cones.

######  Theorem 5.6 ([Per13]).

If for some very ample divisor $H$ on a smooth projective variety $X$ there
exists a transversal covering by $H$-polar $\mathbb{A}^{1}$-cylinders, then
the affine cone over $X$ is flexible.

######  Lemma 5.7.

If $X$ is transversally covered by $\mathbb{A}^{2}$-cylinders, then $X$ is
transversally covered by $\mathbb{A}^{1}$-cylinders. In particular, if for
some very ample divisor $H$ on a smooth projective variety $X$ there exists a
transversal covering by $H$-polar $\mathbb{A}^{2}$-cylinders, then the affine
cone over $X$ is flexible.

###### Proof.

Assume that $X$ has a transversally covering $\\{U_{i}\mid i\in I\\}$, where
each $U_{i}$ is a Zariski open subset in $X$ such that
$U_{i}=\varphi_{i}(\mathbb{A}^{2}\times Z_{i})$. For each $(i,j)\in
I\times\mathbb{A}^{1}$, let $U^{\prime}_{i,j}=U_{i}$,
$Z^{\prime}_{i,j}=Z_{i}\times\mathbb{A}^{1}$, and
$\psi_{j}:\mathbb{A}^{1}\times\mathbb{A}^{1}\to\mathbb{A}^{2},(x,y)\mapsto(x+j,y+j)$.
Then $\varphi^{\prime}_{i,j}:Z^{\prime}_{i,j}\times\mathbb{A}^{1}\to A$ is
defined as follow

---
$\textstyle{Z_{i}\times\mathbb{A}^{1}\times\mathbb{A}^{1}=Z_{i,j}\times\mathbb{A}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\text{id}\times\psi_{j}}$$\scriptstyle{\varphi^{\prime}_{i,j}}$$\textstyle{X}$$\textstyle{Z_{i}\times\mathbb{A}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{i}}$

Therefore $\\{U^{\prime}_{i,j}\mid(i,j)\in I\times\mathbb{A}^{1}\\}$ is
transversally covering by $\mathbb{A}^{1}$-cylinders
$\\{(Z^{\prime}_{i,j},{\varphi^{\prime}}_{i,j})\mid(i,j)\in\mathbb{A}^{1}\\}$
in $X$. ∎

######  Claim 5.8.

Let $X_{2g-2}$ be a general Mukai fourfold of genus $g=7$, $8$ and $9$. Then
there exists an open covering $\\{U_{x}\mid x\in\mathcal{I}\\}$ of $X_{2g-2}$
such that the following conditions hold true.

  1. $i)$

$U_{x}=X_{2g-2}\backslash C_{x}$, where $C_{x}$ is a cubic hypersurface
section of $X_{2g-2}$.

  2. $ii)$

There exists a singular cubic threefold $W_{x}\subset\mathbb{P}^{4}$, and a
birational map $\Phi_{x}:X_{2g-2}\dashrightarrow\mathbb{P}^{4}$ such that

$X_{2g-2}\backslash C_{x}\cong\mathbb{P}^{4}\backslash W_{x}.$

###### Proof.

Let $\mathcal{I}=\\{x\in X\mid\text{ there are exactly five lines in }X\text{
passing through }x\\}$. Then $\mathcal{I}$ is a open set of $X$. By
Proposition 3.2, given a point $x\in\mathcal{I}$, there are a unique cubic
hypersurface section $C_{x}$ of $X_{2g-2}$, a singular cubic threefold
$W_{x}\subset\mathbb{P}^{4}$, and a birational map
$\Phi_{x}:X_{2g-2}\dashrightarrow\mathbb{P}^{4}$ such that

$X_{2g-2}\backslash C_{x}\cong\mathbb{P}^{4}\backslash W_{x}.$

Then we claim that

$\bigcap\limits_{x\in\mathcal{I}}C_{x}=\emptyset$

holds for a general Mukai fourfold $X_{2g-2}$ which implies that $\\{U_{x}\mid
x\in\mathcal{I}\\}$ as above is an open covering. It is sufficient to check
this vanishing condition for one example, since it is an open condition in the
moduli space of Mukai fourfolds of genus $g$. This computation is done in our
ancillary Macaulay2 file (see [HT22]). ∎

######  Theorem 5.9.

The affine cones over general Fano-Mukai fourfold of genus $g=7$, $8$ and $9$
are flexible.

###### Proof.

It follows from Lemma 5.7, Claim 5.8 and Corollary 4.4. ∎

Now, let us consider the generic flexibility of affine cones over Fano-Mukai
fourfolds of genus $g=7$, $8$ and $9$. Then we have the following result.

######  Theorem 5.10.

Let $X_{2g-2}$ be a Fano-Mukai fourfold of genus $g=7$, $8$ and $9$. Then the
affine cone over $X_{2g-2}$ is generically flexible for any polarization of
$X_{2g-2}$.

###### Proof.

Notice that for any ample divisor $H$ on $X_{2g-2}$ all the cylinders
$V_{Q,P}$ are $H$-polar since $\mathrm{Pic}(X_{2g-2})=\mathbb{Z}$. The
statement follows from the above transversal cylinder cover of the complement
of the hypersurface $Y$ in $X_{2g-2}$ and [Per21, Theorem 2.4]. ∎

The affine cones over any smooth Fano-Mukai fourfold of genus $10$ are
flexible ([PrZ20, Theorem A]). An essential ingredient to show the flexibility
in this case was the full description of the Hilbert scheme of cubic surfaces
inside such a Mukai fourfold. A special point in this Hilbert scheme can be
obtained as a special tangential intersection of a Mukai fourfold of genus
$10$, namely, a cone over a twisted cubic (see Section 3.1). To generalize
Theorem 5.10 and 5.9, we conjecture that one also needs descriptions of other
Hilbert schemes of surfaces in Mukai fourfolds of genus $7,8$ and $9$ arising
from special tangential intersections.

######  Problem 5.11.

The affine cones over any smooth Fano-Mukai fourfold of genus
$g\in\\{7,8,9\\}$ are flexible.

## References