Search Results

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 …

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 …

Let $K$ be a non-archimedean field with closed unit disk $\mathcal{O}\subset K$, open unit disk $\mathfrak{m}\subset \mathcal{O}$ and residue field $k = \mathcal{O}/\mathfrak{m}$. Examples to keep in …

Following up on this question which received a negative answer, I wonder if something weaker is true. We work in the same set-up as the previous question. Let $B$ be a smooth quasi-projective variety …

Let $B$ be a smooth quasi-projective variety over a field of characteristic zero. Let $\pi\colon \mathcal{X} \rightarrow B$ be a smooth and projective morphism with geometrically integral fibres. Let …

Section 3 of the following paper of Beauville should answer your question: https://arxiv.org/abs/alg-geom/9701019 In particular, it is shown there that, up to replacing the compactified Jacobian by a …

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 $C$ 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 …

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 …

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 …

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 …

The set $J(C)_{\Theta}[n]$ has the structure of a smooth irreducible algebraic curve, and the restriction of $J(C)\xrightarrow{\times n } J(C)$ to $C$ defines a morphism $J(C)_{\Theta}[n]\rightarrow C …

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 …

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 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^ …