(everything below is defined over an algebraically closed field) 


Let $D$ be a (smooth) surface, and let $X \subset T \times D$ be a flat family of curves on $D$, where $T$ is irreducible. Let $E$ be a curve on $D$. Fix some $t\_0 \in T$ such that $X\_{t\_0}$ does not contain $E$, the intersection $X\_{t\_0} \times\_D E$ is some not necessarily reduced closed subscheme of $D$ of dimension 0.


Is it true then that there exists an open dense subset $U \subset T$, such that $X\_t \times\_D E$ are all isomorphic for different $t \in U$? Does one actually need to assume $D$ to be projective? smooth? Is the previous statement true for an affine $D$?



is intersection of a curve and a family of curves generically constant as a scheme?
Probable counterexample: take $D=\mathbb{A}^2$ (or $\mathbb{P}^2$, if you prefer), $T=\mathbb{A}^1$, and let $X\_t$ be the union of the four lines $x=0$, $y=0$, $x=y$, and $x=ty$. It is well known that $t$ is "almost" determined by the isomorphism class of $D\_t$, as the cross-ratio of the four tangents at the origin. ("Almost" refers to permutations of the four lines, but only finitely many $t$'s give rise to isomorphic $D\_t$'s). My claim is that this remains true if you replace $D\_t$ by a sufficiently big infinitesimal neighbourhood of the origin (presumably the one given by the ideal $(x,y)^5$ works).


Assuming this, you have a counterexample as soon as $E$ contains the neighbourhood in question, i.e. is sufficiently singular at the origin.


