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ag.algebraic geometry – Intersections of strict rework and strict rework of intersections retort

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ag.algebraic geometry – Intersections of strict rework and strict rework of intersections

Let $Z_1,Z_2$ and $Y$ breathe subvarieties of a domestically full intersection selection $X$ over $mathbb C$.

assume the strict transforms of $Z_1$ and $Z_2$ within the blowup $Bl_YX$, the query is: when will their intersection breathe the strict rework of $Z_1 cap Z_2$?

The retort in widespread isn’t any. aircraft if $Y$ is a Cartier divisor, we do not all the time maintain
$overline{Z_1-Y} cap overline{Z_2-Y}= overline{Z_1 cap Z_2-Y}$ in $X$. For occasion, $X=mathbb A^2$ with $Y={y=0}$, $Z_1={x=0}, Z_2={x=y}$.

Another counterexample is $mathbb A^2$ blowing up on the inception $(x,y)$, and $Z_1, Z_2$ are two curves intersecting tangently on the inception.

level to the common property of blow up would not array the unaffected map $ Bl_{Z_1cap Z_2cap Y}(Z_1cap Z_2) rightarrow Bl_{Z_1cap Y}Z_1cap Bl_{Z_2cap Y}Z_2$ is an isomorphism, as a result of the testing objects are restricted. It’s an isomorphism iff restriction of the distinctive divisor to $Bl_{Z_1cap Y}Z_1cap Bl_{Z_2cap Y}Z_2$ is quiet a Cartier divisor (recollect vacant is just too a Cartier divisor).

In drill, one can bridle the query by hand. I’m keen on placing some situations on $Z_1, Z_2, Y$ and $X$ to make the retort correct.

Let me occupy: all $Z_1,Z_2,Y,X$ are equi-dimensional, and $dim(Z_1 cap Z_2)=dim X – dim Z_1 – dim Z_2$; $Z_1 cap Z_2 not subseteq Y$ and $Y not subseteq Z_1 cap Z_2$; $Z_1$ is mild and irreducible; $Bl_Z X$ is smoooth…

The query is (indirectly) associated to: if $f in A$ s.t $barrier{f} in A/I_1, A/I_2$ are each non-torsion, then is $f in A/I_1+I_2$ non-torsion? This is just too pretend in widespread.

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