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| version 1.3, 2000/08/09 03:45:27 | version 1.4, 2000/08/10 02:59:08 | ||
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| % $OpenXM: OpenXM/src/k097/lib/minimal/example-ja.tex,v 1.2 2000/08/02 05:14:30 takayama Exp $ | % $OpenXM: OpenXM/src/k097/lib/minimal/example-ja.tex,v 1.3 2000/08/09 03:45:27 takayama Exp $ | ||
| \documentclass[12pt]{jarticle} | \documentclass[12pt]{jarticle} | ||
| \newtheorem{example}{Example} | \newtheorem{example}{Example} | ||
| \def\pd#1{ \partial_{#1} } | \def\pd#1{ \partial_{#1} } | ||
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| $f=x^3-y^2z^2+y^2+z^2$ $B$H$*$$$?>l9g(B, | $f=x^3-y^2z^2+y^2+z^2$ $B$H$*$$$?>l9g(B, | ||
| $B6u4V(B ${\bf C}^3 \setminus V(f)$ $B$N(B | $B6u4V(B ${\bf C}^3 \setminus V(f)$ $B$N(B | ||
| $B%3%[%b%m%872$N<!85$O(B | $B%3%[%b%m%872$N<!85$O(B | ||
| ${\rm dim}\, H^i = 1$, $(i=0, 1)$, | ${\rm dim}\, H^0 = 8$, ${\rm dim}\, H^1 = 0$, | ||
| ${\rm dim}\, H^i = 0$, $(i=2, 3)$, | ${\rm dim}\, H^2 = 1$, ${\rm dim}\, H^3 = 1$ | ||
| $B$H$J$k(B. | $B$H$J$k(B. | ||
| $B$3$N>l9g(B $D/I$ $B$N(B | $B$3$N>l9g(B $D/I$ $B$N(B | ||
| $b$-$B4X?t$N:GBg@0?t:,$O(B $2$ $B$H$J$j(B, | $b$-$B4X?t$N:GBg@0?t:,$O(B $2$ $B$H$J$j(B, | ||
| $B%3%[%b%m%8$r7W;;$9$k$?$a$K(B | $B%3%[%b%m%8$r7W;;$9$k$?$a$K(B | ||
| $B9M$($k@~7A6u4V$NJ#BN$N<!85$O(B, $10, 12, 9, 4$ $B$G$"$k(B. %%Prog: Srestall.sm1 | $B9M$($k@~7A6u4V$NJ#BN$N<!85$O(B, $20, 28, 27, 11$ $B$G$"$k(B. %%Prog: Srestall_s.sm1 | ||
| $B0lJ}(B Schreyer resolution $B$+$i%9%?!<%H$7$F(B, | $B0lJ}(B Schreyer resolution $B$+$i%9%?!<%H$7$F(B, | ||
| $B@~7A6u4V$NJ#BN$r9M$($k$H(B, $B$=$N<!85$O(B | $B@~7A6u4V$NJ#BN$r9M$($k$H(B, $B$=$N<!85$O(B | ||
| 130, 1078, 1667, 749, 40 $B$H$J$k(B. %%Prog: test21b() | 130, 1078, 1667, 749, 40 $B$H$J$k(B. %%Prog: test21b() | ||
| \end{example} | \end{example} | ||
| $B<!$K(B $(u,v)$-$B6K>.<+M3J,2r$H6K>.<+M3J,2r$,0[$J$kNc$r<($=$&(B. | |||
| \begin{example} \rm | \begin{example} \rm | ||
| %%Prog: minimal-test.k test24() | |||
| $BF1<!2=%o%$%kBe?t$N:8%$%G%"%k(B | |||
| $$I = D^{(h)}\cdot \{ h \pd{x} - x \pd{x} - y \pd{y}, | |||
| h \pd{y} - x \pd{x} - y \pd{y} \} $$ | |||
| $B$r9M$($k(B. | |||
| \begin{tabular}{|l|l|} | |||
| \hline | |||
| Resolution type & Betti numbers \\ \hline | |||
| Schreyer & 1, 3, 3, 1 \\ \hline | |||
| $(-{\bf 1},{\bf 1})$-minimal & 1, 3, 2 \\ \hline | |||
| minimal & 1, 2, 1 \\ | |||
| \hline | |||
| \end{tabular} | |||
| \noindent | |||
| $(-{\bf 1},{\bf 1})$-minimal resolution | |||
| {\footnotesize \begin{verbatim} | |||
| [ | |||
| [ | |||
| [ Dx*h-x*Dx-y*Dy ] | |||
| [ Dy*h-x*Dx-y*Dy ] | |||
| [ x*Dx^2-x*Dx*Dy+y*Dx*Dy-y*Dy^2 ] | |||
| ] | |||
| [ | |||
| [ x*Dx-x*Dy+y*Dy+x*h , -y*Dy-x*h , -h+x ] | |||
| [ -Dy+h , Dx-h , 1 ] | |||
| ] | |||
| ] | |||
| \end{verbatim} | |||
| } \noindent | |||
| $B$G$"$j(B, 1 $BHVL\$N(B syzygy $B$K(B | |||
| \verb# [-Dy+h, Dx-h, 1 ] # | |||
| $B$H(B $1$ $B$,=P8=$7$F$$$k(B. | |||
| $B<+M3J,2r$N<gIt(B (initial) $B$O0J2<$N$H$&$j(B. | |||
| {\footnotesize | |||
| \begin{verbatim} | |||
| [ | |||
| [ | |||
| [ Dx*h ] | |||
| [ Dy*h ] | |||
| [ x*Dx^2-x*Dx*Dy+y*Dx*Dy-y*Dy^2 ] | |||
| ] | |||
| [ | |||
| [ x*Dx-x*Dy+y*Dy , -y*Dy , -h ] | |||
| [ -Dy , Dx , 0 ] | |||
| ] | |||
| ] | |||
| \end{verbatim} | |||
| } | |||
| \noindent | |||
| $B0lJ}(B | |||
| minimal resolution %%Prog: test24b() minimal-test.k | |||
| $B$O(B | |||
| {\footnotesize \begin{verbatim} | |||
| [ | |||
| [ | |||
| [ Dx*h-x*Dx-y*Dy ] | |||
| [ Dy*h-x*Dx-y*Dy ] | |||
| ] | |||
| [ | |||
| [ -Dy*h+x*Dx+y*Dy+h^2 , Dx*h-x*Dx-y*Dy-h^2 ] | |||
| ] | |||
| ] | |||
| \end{verbatim} | |||
| } \noindent | |||
| \end{example} | |||
| \begin{example} \rm | |||
| %Prog: minimal-test.k test20() | %Prog: minimal-test.k test20() | ||
| $I = D\cdot\{ x_1\pd{1}+2x_2\pd{2}+3x_3\pd{3} , | $I = D\cdot\{ x_1\pd{1}+2x_2\pd{2}+3x_3\pd{3} , | ||
| \pd{1}^2-\pd{2}h, | \pd{1}^2-\pd{2}h, | ||
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| \end{minipage} | \end{minipage} | ||
| \end{center} | \end{center} | ||
| $B$A$J$_$K(B, | |||
| $(-w,w)$-$B6K>.<+M3J,2r$H(B $B6K>.<+M3J,2r$,$3$H$J$kNc$r$5$,$7$F$$$k$,(B | |||
| $B$3$l$O$^$@8+$D$+$C$F$$$J$$(B. | |||
| $B$A$g$C$HIT;W5D$G$"$k(B. | |||
| \bigbreak | \bigbreak | ||
| \noindent | |||
| {\tt minimal.k} $B$N%=!<%9%3!<%I$G$O$3$NItJ,$O<!$N$h$&$K$J$C$F$$$k(B. | {\tt minimal.k} $B$N%=!<%9%3!<%I$G$O$3$NItJ,$O<!$N$h$&$K$J$C$F$$$k(B. | ||
| {\footnotesize | {\footnotesize | ||
| \begin{verbatim} | \begin{verbatim} |