Flat sheaves and induced morphism

Question

Hi,

I want to consider a module $P$ on a product $X \times Y$ of varieties over a field of characteristic zero, such that $P$ is flat over $X$. Furthermore I want to consider for each closed point $x$ on $X$ the natural map

$i_x:Y\rightarrow X\times Y$,

which comes from the cartesian diagram induced by the inclusion of $x$ in $X$, so naively it is just: sending a point $y$ on $Y$ to $(x,y)$ for that fixed $x$. With $P_x$ I want to denote the pullback of $P$ via $i_x$.

Assume that for every closed $x$ you have that $P_x$ is the skyscraper $k(y)$ for a closed point $y$ on $Y$. Then I can define a map of sets

$f:Cl(X)\rightarrow Cl(Y)$

between the closed points $Cl(X)$ and $Cl(Y)$ on $X$ and $Y$. My question: can you use this $f$ to define a real morphism between $X$ and $Y$ as varieties?

Remark: in the case I am interested in the f is bijective. Does this then imply that, if the extension exists, it is automatically an iso?

Thanks!

Answer 1

I will assume that $X$ is normal. Let $Z$ be the support of $P$ on $X\times Y$. Then by your assumption $Z$ projects bijectively onto $X$. Since we are in characteristic $0$, this map is separable, so by Zariski's main theorem, this is an isomorphism. Then composing the inverse with this isomorphism with the projection to $Y$ you get a morphism $X\to Y$.

Answer 2

I saw this question while trying to understand the proof of Cor. 5.23 of Huybrecht's Furier-Mukai transform. With exactly the same assumptions, there is a conclusion that using local sections of the module one can define a morphism. This cannot be difficult, but I am in the middle of understanding it. There is also an answer to the second question: if varieties are smooth then it is an iso. If not Huybrechts uses the assumption that derived categories are equivalent.