aaaaaaaaa $\mathit{a}$ $\mathit{a}$ $\mathit{a}$ above, along along also an and and and and and and and and

and and and and another as At

be be be be be be because by by by by by

$\mathrm{\setminus }\mathrm{.}\mathrm{a}\mathrm{s}\mathrm{e}$ case case case, classifying CM CM CM cokernel cokernel common common complete

$\mathrm{\wedge }\mathrm{.}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{e}$ construction COROLLARY corresponding

Jeduce define define define defined defines defines degrees double

fxact exact exists

['or for form freely Further,

generate generated given global graded

handle has has have have have have hence hence hence hence holds. homogeneous

homomorphism

tdeal ideal ideal ideal If If If If if if images immediately In In in in inclusion

${\mathit{j}}_{\mathit{n}\mathit{f}\mathit{i}\mathit{n}\mathit{i}\mathit{t}\mathit{e}\mathit{s}\mathrm{i}\mathit{m}\mathit{a}\mathit{l}}$ intersection intersection is is is is is is is is is is is is

kernel kernel Koszul

Latter lemma length, Let Let Let let Letting line line line lines lines

map map multiplicity multiplicity multiplicity multiplicity multiplicity must

$\mathrm{b}\mathrm{-}\mathrm{1}\mathrm{=}\mathrm{deg}\mathit{p}$ $\mathrm{=}\mathrm{0}\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}{{O}}_{\mathit{W}}\mathrm{\ge }\mathrm{1}\mathrm{,}$ ${\mathit{H}}_{\mathrm{*}}^{\mathrm{1}}\mathrm{\left(}{\mathrm{I}}_{\mathit{Z}}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathrm{,}$ ${\mathit{p}}_{\mathit{a}}\mathrm{\left(}\mathit{W}\mathrm{\right)}\mathrm{=}\mathrm{1}\mathrm{-}\mathit{b}$. ${\mathit{p}}^{\mathrm{\prime }}\mathrm{=}\mathit{c}\mathit{p}\mathrm{m}\mathrm{o}\mathrm{d}{\mathit{I}}_{\mathit{Y}}$

${\mathit{q}}^{\mathrm{\prime }}\mathrm{=}\mathit{c}\mathit{q}\mathrm{m}\mathrm{o}\mathrm{d}{\mathit{I}}_{\mathit{Y}}$. ${\mathit{S}}_{\mathit{Y}}$-submodule $\mathit{u}$ : ${\mathrm{I}}_{\mathit{Z}}\mathrm{\to }{{O}}_{\mathit{Y}}\mathrm{(}\mathit{b}\mathrm{-}\mathrm{2}\mathrm{)}\mathit{u}$ : ${\mathrm{I}}_{\mathit{Z}}\mathrm{\to }{{O}}_{\mathit{Y}}\mathrm{(}\mathit{b}\mathrm{-}\mathrm{}\mathrm{2}\mathrm{)}$. ${\mathit{S}}_{\mathit{Y}}\mathrm{(}\mathrm{-}\mathrm{1}\mathrm{)}\mathrm{\oplus }{\mathit{S}}_{\mathit{Y}}\mathrm{(}\mathrm{-}\mathrm{2}\mathrm{)}$,

$\mathit{I}{\mathit{i}}_{\mathit{Z}}\mathrm{/}{\mathit{I}}_{\mathit{Z}}{\mathit{I}}_{\mathit{Y}}\mathrm{\to }{\mathit{S}}_{\mathit{Y}}\mathrm{(}\mathit{b}\mathrm{-}\mathrm{2}\mathrm{)}\mathrm{(}{\mathit{I}}_{\mathit{Y}}^{\mathrm{2}}\mathrm{,}\mathrm{}\mathit{x}\mathit{q}\mathrm{)}\mathrm{\subset }\mathrm{(}{\mathit{y}}^{\mathrm{2}}\mathrm{,}\mathrm{}\mathit{x}\mathrm{)}\mathrm{0}\mathrm{\to }{\mathrm{I}}_{\mathit{W}}\mathrm{\to }{\mathrm{I}}_{\mathit{Z}}\mathrm{\to }{{O}}_{\mathit{Y}}\mathrm{(}\mathit{b}\mathrm{-}\mathrm{}\mathrm{2}\mathrm{)}$ $\mathrm{\to }\mathrm{0}$.

$\mathrm{3}\mathrm{\to }{{O}}_{\mathit{Y}}\mathrm{(}\mathit{b}\mathrm{-}\mathrm{2}\mathrm{)}\mathrm{\to }{{O}}_{\mathit{W}}\mathrm{\to }{{O}}_{\mathit{Z}}\mathrm{\to }\mathrm{0}$,

$\mathit{b}\mathit{p}\mathit{p}\mathit{p}\mathit{\phi }\mathit{q}\mathit{q}\mathit{q}\mathit{u}\mathit{W}\mathit{W}\mathit{W}\mathit{W}\mathit{W}\mathit{W}\mathit{W}\mathit{x}\mathit{Y}\mathit{Y}\mathit{Y}\mathit{Y}\mathit{Z}\mathit{Z}\mathit{Z}\mathit{Z}\mathit{Z}\mathit{p}\mathrm{,}$ ${\mathit{q}}^{\mathrm{\prime }}{\mathit{q}}^{\mathrm{\prime }}\mathit{W}\mathrm{,}$ $\mathit{Y}$,

Y. Y. Y. Y. $\mathit{Z}\mathrm{,}$ $\mathit{b}\mathrm{=}\mathrm{1}{\mathit{p}}^{\mathrm{\prime }}\mathrm{,}$ ${\mathit{p}}^{\mathrm{\prime }}\mathrm{,}$ ${\mathit{S}}_{\mathit{Y}}\mathrm{,}$ ${\mathit{W}}^{\mathrm{\prime }}\mathrm{,}$ ${\mathit{y}}^{\mathrm{2}}$. $\mathit{a}\mathrm{=}\mathrm{-}\mathrm{1}\mathit{a}\mathrm{>}\mathrm{0}$. $\mathit{b}\mathrm{\ge }\mathrm{0}\mathrm{:}\mathit{b}\mathrm{-}\mathrm{1}\mathrm{,}$ $\mathit{c}\mathrm{\in }{\mathit{k}}^{\mathrm{*}}\mathit{q}\mathrm{=}\mathrm{0}$,

$\mathit{q}\mathrm{=}\mathrm{0}\mathrm{,}$ $\mathit{q}\mathrm{\ne }\mathrm{0}\mathrm{,}$ $\mathit{q}\mathrm{\ne }\mathrm{0}\mathrm{,}$ $\mathit{W}\mathrm{=}{\mathit{W}}^{\mathrm{\prime }}\mathit{x}\mathrm{\mapsto }\mathit{p}\mathrm{,}$ $\mathit{Y}\mathrm{\subset }{\mathbb{P}}^{\mathrm{3}}\mathit{a}\mathrm{=}\mathrm{-}\mathrm{1}\mathrm{,}$ $\mathit{a}\mathrm{=}\mathrm{-}\mathrm{1}\mathrm{,}$ $\mathrm{(}\mathit{q}\mathrm{,}\mathit{p}\mathrm{)}\mathit{x}\mathrm{\mapsto }\overline{\mathit{p}}\mathrm{,}$ ${\mathit{y}}^{\mathrm{2}}\mathrm{\mapsto }\overline{\mathit{q}}$

${\mathit{y}}^{\mathrm{2}}\mathrm{\mapsto }\mathit{q}$. ${\mathit{I}}_{\mathit{Z}}\mathrm{\otimes }{\mathit{S}}_{{\mathrm{1}}^{\mathrm{\Gamma }}}\mathrm{=}\mathrm{(}\mathit{a}\mathrm{,}\mathrm{}\mathit{b}\mathrm{)}\mathrm{,}$ $\mathrm{(}\mathrm{-}\mathrm{1}\mathrm{,}\mathrm{}\mathit{b}\mathrm{)}{\mathrm{I}}_{\mathit{W}}\mathrm{=}\mathrm{k}\mathrm{e}\mathrm{r}\mathit{u}{\mathit{I}}_{\mathit{W}}\mathrm{=}{\mathit{I}}_{\mathit{Y}}^{\mathrm{2}}\mathrm{,}$ $\mathrm{(}\overline{\mathit{p}}\mathrm{,}\mathrm{}\overline{\mathit{q}}\mathrm{)}\mathrm{(}{\mathit{q}}^{\mathrm{\prime }}\mathrm{,}{\mathit{p}}^{\mathrm{\prime }}\mathrm{)}\mathrm{(}\mathit{x}\mathrm{,}\mathrm{}{\mathit{y}}^{\mathrm{2}}\mathrm{)}\mathrm{,}$ $\mathrm{(}{\mathit{y}}^{\mathrm{2}}\mathrm{,}\mathrm{}\mathit{x}\mathrm{)}$,

$\mathrm{[}\mathit{x}\mathrm{=}\mathit{y}\mathrm{=}\mathrm{0}\mathrm{\}}\mathit{W}\mathrm{=}{\mathit{Y}}^{\mathrm{\left(}\mathrm{2}\mathrm{\right)}}$. $\mathit{W}\mathrm{=}{\mathit{Y}}^{\mathrm{\left(}\mathrm{2}\mathrm{\right)}}$. ${\mathit{H}}_{\mathrm{*}}^{\mathrm{0}}\mathrm{\left(}\mathit{u}\mathrm{\right)}\mathrm{=}\mathit{\phi }\mathrm{\{}\mathit{x}\mathrm{=}{\mathit{y}}^{\mathrm{2}}\mathrm{=}\mathrm{0}\mathrm{\}}{\mathit{p}}_{\mathit{a}}\mathrm{\left(}\mathit{W}\mathrm{\right)}\mathrm{=}\mathrm{1}\mathrm{-}\mathit{b}\mathit{x}\overline{\mathit{q}}\mathrm{-}{\mathit{y}}^{\mathrm{2}}\overline{\mathit{p}}$,

${\mathit{r}}_{\mathit{W}}\mathrm{=}\mathrm{(}{\mathit{x}}^{\mathrm{2}}\mathrm{,}\mathrm{}\mathit{x}\mathit{y}\mathrm{,}\mathrm{}{\mathit{y}}^{\mathrm{3}}\mathrm{,}\mathrm{}\mathit{x}\mathit{q}\mathrm{-}{\mathit{y}}^{\mathrm{2}}\mathit{p}\mathrm{)}{\mathit{I}}_{\mathit{Z}}\mathrm{/}{\mathit{I}}_{\mathit{Z}}{\mathit{I}}_{\mathit{Y}}\mathrm{\cong }{\mathit{S}}_{\mathit{Y}}\mathrm{(}\mathrm{-}\mathrm{1}\mathrm{)}\mathrm{\oplus }{\mathit{S}}_{\mathit{Y}}\mathrm{(}\mathrm{-}\mathrm{2}\mathrm{)}\mathit{\phi }$ : ${\mathit{I}}_{\mathit{Z}}\mathrm{\to }{\mathit{I}}_{\mathit{Z}}\mathrm{/}{\mathit{I}}_{\mathit{Z}}{\mathit{I}}_{\mathit{Y}}\mathrm{\to }{\mathit{S}}_{\mathit{Y}}\mathrm{(}\mathit{b}\mathrm{-}\mathrm{}\mathrm{2}\mathrm{)}$.

${\mathit{H}}_{\mathrm{*}}^{\mathrm{1}}\mathrm{\left(}{\mathrm{I}}_{\mathit{W}}\mathrm{\right)}\mathrm{\cong }\mathrm{(}\mathit{S}\mathrm{/}\mathrm{(}\mathit{x}\mathrm{,}\mathrm{}\mathit{y}\mathrm{,}\mathit{p}\mathrm{,}\mathrm{}\mathit{q}\mathrm{\left)}\mathrm{\right)}\mathrm{(}\mathit{b}\mathrm{-}\mathrm{}\mathrm{2}\mathrm{)}$

${\mathit{r}}_{\mathit{W}}\mathrm{+}{\mathit{I}}_{\mathit{Y}}^{\mathrm{2}}\mathrm{=}\mathrm{(}{\mathit{x}}^{\mathrm{2}}\mathrm{,}\mathrm{}\mathit{x}\mathit{y}\mathrm{,}\mathrm{}{\mathit{y}}^{\mathrm{2}}\mathrm{,}\mathrm{}\mathit{x}\mathit{q}\mathrm{)}\mathrm{=}\mathrm{(}{\mathit{I}}_{\mathit{Y}}^{\mathrm{2}}\mathrm{,}\mathrm{}\mathit{x}\mathit{q}\mathrm{)}$. $\mathrm{k}\mathrm{e}\mathrm{r}\mathit{\phi }\mathrm{=}\mathrm{(}{\mathit{I}}_{\mathit{Z}}{\mathit{I}}_{\mathit{Y}}\mathrm{,}\mathrm{}\mathit{x}\mathit{q}\mathrm{-}{\mathit{y}}^{\mathrm{2}}\mathit{p}\mathrm{)}\mathrm{=}\mathrm{(}{\mathit{x}}^{\mathrm{2}}\mathrm{,}\mathrm{}\mathit{x}\mathit{y}\mathrm{,}\mathrm{}{\mathit{y}}^{\mathrm{3}}\mathrm{,}\mathrm{}\mathit{x}\mathit{q}\mathrm{-}{\mathit{y}}^{\mathrm{2}}\mathit{p}\mathrm{)}$.

$\mathrm{\cap }\mathrm{.}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}\mathit{\phi }\mathrm{=}{\mathit{S}}_{\mathit{Y}}\mathrm{(}\mathit{b}\mathrm{-}\mathrm{2}\mathrm{)}\mathrm{/}\mathrm{(}\overline{\mathit{p}}\mathrm{,}\mathrm{}\overline{\mathit{q}}\mathrm{)}\mathrm{\cong }\mathrm{(}\mathit{S}\mathrm{/}\mathrm{(}\mathit{x}\mathrm{,}\mathrm{}\mathit{y}\mathrm{,}\mathit{p}\mathrm{,}\mathrm{}\mathit{q}\mathrm{\left)}\mathrm{\right)}\mathrm{(}\mathit{b}\mathrm{-}\mathrm{2}\mathrm{)}$

${\mathrm{v}}^{\mathrm{O}}$ no NOLLET not

af of of of of of of of of of on on on on on on on only or otherwise

$\mathrm{L}\mathrm{9}\mathrm{9}\mathrm{7}\mathrm{2}\mathrm{.}\mathrm{1}\mathrm{.}\mathrm{2}\mathrm{.}\mathrm{2}\mathrm{.}\mathrm{3}\mathrm{3}\mathrm{0}\mathrm{3}\mathrm{7}\mathrm{4}{\mathrm{4}}^{\mathrm{e}}$ filtrant filtrant finite finite (a) (b) (b) (c) (c) (c). (d) (d) (e) (e).

part part points polynomials polynomials principal Proof. properties property PROPOSITION

provides proving

quasiprimitive quasiprimitive

regular relation

$\mathrm{\neg }\mathrm{,}\mathrm{a}\mathrm{m}\mathrm{e}$ second second second second see see see separately. sequence sequence sequence SERIE

sheaf sheafifies shows simply Since Since Since Since snake so $\mathrm{s}$. structure structure

structure structure structure structure subscheme such supported support. surjection surjection

that that that that that that that The The The the the the the the the the the the the the

the the the the the the the The the the the the then then Then then there This this this

this three three three three to to TOME total triple Triple two two two type tyPe type

unit, unless use

we we we we we we we when whence where which which which while will will with

with with

ieros zeros