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No. Let $S = \mathbb{A}^1_{\mathbb{C}}$ and $\mathcal{X} = \mathbb{A}^1_S \rightarrow S$ be the constant family. Let $\mathcal{Y}$ be the blow-up of the surface $\mathbb{P}^1_S$ at the closed point ov …

Here's a suggestion (not a full answer): take a geometric point $\bar{s} \rightarrow s$ above $s$. Then the stalk $\left(R^1 j_* P_{\eta}\right)_{\bar{s}}$ is computed as the Galois cohomology $H^1(K^ …

This is true, and can be shown by an induction argument on $h^0(A)$. If $h^0(A)=2$, then $\deg(A)\geq 4$ since the gonality of $X$ is $4$. If $h^0(A)>2$, let $p\in X$ be a point in the support of an e …

This is false. Let $G = \mathrm{GL}_2$ and $X$ be the vector space of binary quartic forms $q \in \mathbb{C}[x,y]$, with action given by linearly substituting and dividing by the square of the determi …

Suppose $X/\mathbb{Q}$ is a (smooth, projective, geometrically integral) curve of genus $g\geq 2$ and $J/\mathbb{Q}$ its Jacobian variety. If one is interested in determining the (finite, by Faltings) …

This already fails for line bundles on smooth projective curves: let $X$ be 'the' pointless conic over $\mathbb{R}$, given by the closed subscheme of $\mathbb{P}^2_{\mathbb{R}}$ cut out by $X^2+Y^2+Z^ …

Let $X$ be a quasi-projective scheme over a field $k$. Let $G$ be a finite group acting on $X$ whose order is invertible in $k$. If $X$ is Cohen-Macaulay, can we conclude that the subscheme of fixed p …

The $n$-dimensional weighted projective space $X = \mathbb{P}(k_1,\dots,k_{n+1})$ is a toric variety, hence is automatically rational since it contains an isomorphic copy of $\mathbb{G}_m^n$ as a dens …

Let $C$ be a reduced, connected, projective and purely one-dimensional scheme of finite type over a field $k$. Suppose that $C$ is rational, i.e. that the normalisation of $C$ is a disjoint union of c …

Yes. This follows from Theorem 1.10(ii) of the paper of Altman-Kleiman cited below. More precisely, let $S$ be a scheme and let $\mathcal{F}$ be a locally finitely presented $\mathcal{O}_{X_S}$-module …

This is true and follows from: Claim: Let $x$ be a $n\times n$ matrix with $\mathbb{C}$-coefficients. Then the centralizer $C(x)$ of $x$ in $GL_n(\mathbb{C})$ fits into a short exact sequence $1\right …

Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. Let $\pi\colon X\rightarrow \text{Spec}(R)$ be a smooth projective morphism with geometrically integral fibers. In F …

In case this might be useful to anyone, it turns out that results of Colliot--Thelene can be used to resolve a closely related question: see Theorem 6.1 in the following preprint (apologies for the se …

Let $k$ be a characteristic zero field, $V \subset \mathbb{A}^n_k$ an open subscheme, $G$ a split reductive group over $k$ and $T$ a $G$-torsor over $V$ (in the etale, equivalently fppf topology). Sup …

One possible answer to this could be Lang's theorem: it says that if $G/\mathbb{F}_q$ is a smooth connected algebraic group, then $H^1(\mathbb{F}_q,G)$ is trivial, or otherwise put every $G$-torsor ha …