I am encountering the following question now.


Consider a smooth and irreducible projective variety over complex numbers. Let $L$ be an ample, globally generated line bundle on $X$. For $r$ satisfying $2\leq r\leq h^0(X,L)$ consider general $r$ dimension subspaces $V\subset H^0(X,L)$.


Consider the zero locus of these spaces, these are closed subschemes $Z\_V$ whose ideals are the images of $V\otimes L^{-1}\rightarrow O\_X$. 


Since we choose $V$ to be general enough the closed subschemes $Z\_V$ have codimension exactly $r$ in $X$.


I am interested in understanding whether these closed subschemes move in a family. More precisely if we take $V$ and $V'$ general $r$ dimensional subspaces, then will $Z\_V$ and $Z\_{V'}$ have the same Hilbert polynomial?


What is the Hilbert polynomial of such a $Z\_V$.


If $Z\_V$'s have the same Hilbert polynomial, then so will their ideal sheaves. So they will also move in a flat family.


Thanks in advance!



Closed subschemes defined by sections of a line bundle
I believe the answer is yes. If your $V$ is spanned by the sections $s\_1,\dots,s\_k$, then locally they form a regular sequence, so you can form a Koszul complex, which induces an exact sequence
$$
0\to L^{\otimes(-r)}\to(L^{\otimes(-r+1)})^r\to (L^{\otimes(-k+2)})^{r(r-1)/2}\to \dots\to (L^{-1})^r\to \mathcal I\to 0,
$$
where $\mathcal I$ is the ideal sheaf of $Z\_V$. It is obvious from this exact sequence that Hilbert polynomial of $\mathcal I\_V$ is an alternating sum of Hilbert polynomials of the other terms of the sequence, which is independent of the coice of sections provided they cut out a subscheme of codimension $r$.


