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As for a result that was not simply incorrectly proved, but actually false, there is the case of the Severi bound(*) for the maximum number of singular double points of a surface in P^3. The predicti …

[EDIT: A previous version mistakenly argued that the fundamental group of X was responsible for torsion in the Picard group. I hope that this is correct now! Btw, there is probably a more direct way …

A great difference in the transition from varieties to schemes is the presence of non-reducedness. Sometimes on a given scheme there are different natural scheme structures and with respect to one, t …

A result of Kawamata (Kawamata, Yujiro, Characterization of abelian varieties. Compositio Mathematica, 43 no. 2 (1981), p. 253-276) implies that, under your assumptions, $X$ is birational to an abelia …

A few opposite-looking remarks, and the case of (minimal) surfaces. If a variety $X$ admits a non-constant morphism to a curve, then it admits a non-constant morphism to $\mathbb{P}^1$ (just compose …

I would like to mention one more homotopy invariant of smooth projective varieties that is also a birational invariant. If $X$ is a smooth projective variety, then the torsion subgroup $T(X)$ of ${\rm …

Let me expand on my comment. Let $E$ be an elliptic curve and let $p$ be a point of $E$ of infinite order. Embed $E$ in $\mathbb{P}^2$ as a plane cubic using the linear system $3O$, where $O$ is the …

I suspect that even if you had a single curve over $\mathbb{F}_p$, you might not find a lift to $\mathbb{Q}$. Below I sketch an argument that works under the assumption that $\mathcal{M}_g$ does not …

I do not know about the local-to-global principle for testing rationality of a surface, but weakening "rational" to "unirational" it certainly fails. Indeed, del Pezzo surfaces violating the Hasse pr …

It is certainly not irreducible if n=8 and d>3. This is analyzed nicely in the paper Hilbert schemes of 8 points, Dustin A. Cartwright, Daniel Erman, Mauricio Velasco, Bianca Viray available at htt …

I am indeed claiming that it works over any field, but with the additional assumption that the morphism f be separable (this is required to make the induced morphism between the normalizations an isom …

Here is one easy case of a curve of genus two whose Jacobian is isogenous to a product of two elliptic curves. Let $p,q,r$ be homogeneous separable polynomials of degree two in two variables $s,t$ th …

I am interpreting the question as also implying that $X$ itself is not a product. I do not have an answer in general, but I think that I have an example. It is a projective surface with a singular p …

It seems, that in the case in which E and F are direct sums of line bundles (and G is non-zero!), you can reconstruct E and F knowing that $E \otimes G \simeq F \otimes G$: this simply imitates the pr …

This is a substantial revision of my previous answer based on the comments and the other answers. I am making it community wiki for two reasons: it incorporates ideas from my answer (for which I alrea …