TITLE: Line bundle on a product
QUESTION [6 upvotes]: Let X and T be projective varieties, with $H^1(\mathcal{O}_X)=0$. Take L a line bundle on the product. Prove that for any two points $t,t'$ of T, the pullbacks $L_t, L_{t'}$ to $X\times t, X\times t'$ are isomorphic line bundles on X.
I am completely stuck and I don't even undeerstand how to use  the hypothesis in cohomology. If someone posts an hint I can try to elaborate.

REPLY [3 votes]: Let $X$ be an integral projective scheme over an algebraically closed field $k$, and assume that $H^1(X, \mathcal{O}_X) = 0$. Let $T$ be a connected scheme of finite type over $k$. We want to show that if $\mathscr{L}$ is an invertible sheaf on $X \times T$, then the invertible sheaves $\mathscr{L}_t$ on $X = X \times \{t\}$ are isomorphic, for all closed points $t \in T$. This is the content of Exercise III.12.6(a) of Hartshorne, as mentioned in the comments above.
We have a (group) scheme $\text{Pic}(X)$ which essentially parametrizes line bundles on $X$ (and which happens to be an infinite disjoint union of projective schemes, although that's not terribly relevant). $H^1(X, \mathcal{O}_X)$ is the tangent space of $\text{Pic}(X)$ (at every point), so if it is $0$, it follows that $\text{Pic}(X)$ is actually a disjoint union of (reduced) points. Then $\mathscr{L}$ induces a morphism $T$ to $\text{Pic}(X)$, which must be constant since $T$ is connected and $\text{Pic}(X)$ is a disjoint union of points. Because the morphism $T$ to $\text{Pic}(X)$ is constant, the fibers $\mathscr{L}_t$ are all isomorphism (we have to be slightly careful — morphisms $T$ to $\text{Pic}(X)$ are not in bijection with line bundles of $X \times T$, but any two line bundles inducing the same morphism will have isomorphic fibers $\mathscr{L}_t$). For details on this material, look at the book "Néron models" here.
Bosch, Siegfried; Lütkebohmert, Werner; Raynaud, Michel. Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 21. Springer-Verlag, Berlin, 1990. x+325 pp.