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The splitting theorem is most certainly false for vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$. In fact, the theory of vector bundles on quadric surfaces is probably as complicated as the theory …

I don't think this is true, even for stable bundles. Assume that the genus of $X$ is at least 2. Take a point $p$ of $X$, and a nontrivial extension of ${\cal O}(p)$ by ${\cal O}$, this is indecomposa …

Let $X$ be the scheme obtained by gluing the generic points of all $\operatorname{Spec}\mathcal{O}_p$ for all closed points $p$ of $\mathbb{A}^1_{\mathbb C}$. The obvious morphism $X \to \mathbb{A}^1_ …

Check out Section 2 of Noohi's paper Fundamental groups of topological stacks with slice property, Algebr. Geom. Topol. 8 (2008) pp 1333–1370, doi:10.2140/agt.2008.8.1333, arXiv:0710.2615.

Tame Artin stacks (in the sense of Abramovich, Olsson and Vistoli, https://math.berkeley.edu/~molsson/tame.pdf) with quasi-projective moduli spaces will have property 2: the line bundle is the pullba …

The map $U \to \mathbb{G}(k,n)$ that you are interested in is flat, and a flat finitely presented map is well known to be open. Flatness can be checked in several ways: for example, the map is generi …

Your conjecture 2 is false if you don't assume that $G$ is reduced (in positive characteristic there are affine group schemes that are not reduced). As to conjecture 3, it is hopelessly wrong (think …

The Betti numbers of many (complex) moduli spaces have been computed by counting points over finite fields, using the Weil conjectures, as proved by Deligne, and comparison theorems for étale and sing …

Actions of reductive groups with finite stabilizers on quasi-projective varieties are often not proper. The simplest example I know is given by he action of $\mathrm{PGL}_2$ on the projective space $\ …

If $A_1$, $A_2$ and $A_3$ are in distinct fibers of the double cover $C \to \mathbb P^1$, then $\mathrm h^0(C, L) = 1$. Otherwise you would get a map $C \to \mathbb P^1$ of degree 3; together with the …

Suppose $G$ is an affine group scheme over an algebraically closed field $k$, and $V$ is a finite dimensional representation of $G$. Let $K$ be an extension of $k$, and assume that $V_K$ is reducible …

Of course, we all know great mathematicians who constantly make mistakes even now, and not because of foundations. In any case, it's not like "long dead Italian algebraic geometers" is a category of …

If $Y$ is a complex projective algebraic variety, the Picard group $\operatorname{Pic}Y$ has the structure of an algebraic variety; if $X$ is another algebraic variety, any line bundle $L$ gives a reg …

The stack of curves of genus 0 with at most one node is a quotient stack (see The integral Chow ring of the stack of at most 1-nodal rational curves, but Edidin and Fulghesu), so you are fine in this …

The notion of anti-analytic involution is perfectly well defined for general analytic spaces: it is an involution of locally ringed spaces, that is antilinear with respect to complex scalars. Any pro …