# Tagged Questions

The first purpose of schemes theory is the geometrical study of solutions of algebraic systems of equations, not only over the real/complex numbers, but also over integer numbers (and more generally over any commutative ring with 1).It was finalized by Alexandre GROTHENDIECK, during the 1950's and ...

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### Useful, non-trivial general theorems about morphisms of schemes

I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians. I'm trying to ...
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### Which locally ringed spaces are schemifiable?

(most of this question is re-asking Schemification (schematization?) of locally ringed spaces, which did not get answered) Given a locally ringed space $X$, say that a schemification of $X$ is a ...
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### Why define curves over perfect fields?

One may define a curve (e.g. separated scheme of finite type of dim. 1) over an algebraically closed field, as done in Hartshorne's book. A weaker assumption, which is used commonly, is to define a ...
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### Action of a lattice on abelian varieties

Let $\pi\colon Y\to\mathbb{P}_\mathbb{C}^1$ be a ramified cover of degree two of $\mathbb{P}_{\mathbb{C}}^1$ such that $Y$ is smooth. I fix a point $x$ on $\mathbb{P}^1$, over which the cover is étale,...
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### functions coming from a perverse sheaf

Let's take a scheme $X$ over a finite field $k$ and $f:X(k)\rightarrow\mathbb{Q}_{\ell}$ What kind of condition do I need on $f$ if I want that it comes from an irreducible perverse sheaf on $X$?
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### Some questions about ruled surfaces defined over a number field

definitions: A non-singular complex projective surface $S$ is a ruled surface if it is birationally equivalent to $C\times_{\text{Spec} \mathbb C}\mathbb P^1_{\mathbb C}$ where $C$ is a non-singular ...
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### Action on algebraic variety and adjoint bundles

Let $X$ be a complex algebraic variety and let $G$ be a complex algebraic group; I mean that $X$ is a reduced, separated scheme of finite type on $\operatorname{Spec}\mathbb{C}$, and the underlying ...
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### representability of some mapping stack

Let $S$ be an Artin stack of finite type. We assume that it contains a point as an open dense. Is it always true that the mapping stack: $Hom^{0}(\mathbb{P}^{1},S)$ which consists of sections ...
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### When is the determinant of the push-forward of an ample line bundle ample

Let $f:X\to S$ be a "nice" morphism of "nice" schemes. Let $L$ be an ample line bundle on $X$. When is $\det f_\ast L$ also ample? A "nice" morphism could be anything from "finite type separated" to ...
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### Verdier duality on excellent schemes

Let $f:X\rightarrow Y$ be a regular morphism between $k$-schemes which are noetherian and excellent with a funcion of dimension. In the book by Illusie-Laszlo-Orgogozo, there is a theorem (4.4.1 in ...
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### Obstructed automorphisms of schemes

Let $X$ be a smooth projective scheme over a field $\mathbf{k}$ of characteristic zero such that $\mathrm{H}^0(X, \mathrm{T}X)$ vanishes, and let $f$ be an automorphism of $X$. I would like to have an ...
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### extending local systems on a neighbourhood

Let $Y$ an affine finite type scheme over an algebraically closed field $k$. Let $S$ be a closed subscheme of $Y$ and $Y'$ the henselization of $Y$ along $S$. If we have a $\mathbb{Z}_{\ell}$ local ...
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### l-adic local system. on hensel schemes

Let $k$ be a field, $\ell$ a prime different from the characteristic. If I take $S$ a closed subscheme of $Y$, which is a $k$-scheme of finite type, is it true that any $\mathbb{Z}_{\ell}$-local ...
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### Higher-dimensional Artin L-functions

I begin by clarifying that the "higher-dimensional" in my question refers to analogues of Artin L-functions over higher dimensional base schemes than $\mathrm{Spec}(\mathbb{Z})$. Now for the set-up. ...
Let $X$ be a scheme over $\mathbb{C}$. When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex? When does the topological ...