Questions tagged [schemes]
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 1950s and the 1960s.
810 questions
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Dimension formula for the pullback of smooth affine schemes along a retraction
Let $A, B$ and $C$ be smooth affine schemes.
Let $r:A \to B$ be a retraction (i.e. there is a right-inverse $s:B \to A$ such that $r \circ s = \mathrm{id}_B$). Let $f: C \to B$ be a morphism that is ...
- 131
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Can every variety be embedded into a toric variety?
Let $V$ be a complete (integral) variety over a field $k$. I am happy to assume that $k$ is algebraically closed.
Is $V$ always embeddable into some toric variety $T$ (edit: as a closed subscheme)? If ...
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154
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Decomposition of $G$-equivariant locally free sheaf
Let $X$ be an irreducible $k$-scheme of finite type, $G$ a finite group of order coprime to $\text{char}(k)$, $\mathcal{F}$ a locally free $\mathcal{O}_X$-module admitting additionally structure of $G$...
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Extending the "infinitesimal automorphism" construction of derivations to algebraic geometry
In the context of smooth manifolds $M$, there is a well-known correspondence between the "infinitesimal" versions of certain objects and derivations of the algebra $C^\infty(M, \mathbb{R})$. ...
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Irreducibility of fibres of dominant morphism with equidimensional fibres
Let $f: X \to Y =\operatorname{Spec}(A)$ a dominant morphism between irreducible smooth varieties (= $k$-schemes of finite type) such that
the generic fibre $X_{\eta}$ is irreducible
every fibre $X_y$...
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Descent theory and global sections
Let $g: S' \to S$ be a quasi-compact faithfully flat morphism and let
$\text{pr}_i : S' \times_S S' \to S'$ & $\text{pr}_{ij} : S' \times_S S' \times_S S' \to S' \times_S S'$ ($i=1,2,3$) the ...
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Properties of orbit under group scheme action & dimension count formula
Let $X/k$ be a scheme over base field $k$ and $G$ an group scheme (also over $k$) acting on $X$ (ie there is an algebraic map $\sigma: G \times X \to X$ satisfying some compatibility conditions).
Let $...
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A question related to Stein factorisation
I am trying to prove this theorem,
Let $X=\prod_{i=1}^n X_i $.
Theorem: Let $f: X \rightarrow X_i$ is a projection map (i.e., surjective with connected fibers) and $ g: X\rightarrow Z$ is a proper ...
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Representability criterion for functors coarsely represented
Consider the following representability criterion for functors:
Let $F:(\mathrm{Sch}/S)^{\mathrm{op}}\to\mathbf{Sets}$ be a moduli functor. Suppose:
(i) $F$ is a sheaf for the Zariski topology;
(ii) ...
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K-polystable Fano varieties
I am trying to understand the proof by Chen, Donaldson and Sun of the YTD conjecture, i.e.
"A Fano variety is K-polystable if and only if it admits a K"ahler-Einstein metric".
The ...
- 119
3
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1
answer
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Definition of Adelic points of a scheme over $\mathbb{Q}$
In Milne's notes on Shimura Varieties (https://www.jmilne.org/math/xnotes/svi.pdf) at the start of section 4, it defines the ring of finite adeles. Then for an affine variety $V=\mathrm{Spm}(A)$ over $...
- 129
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Dimension of Scheme bounds dimension of Proj space $ \operatorname{Proj}(\bigoplus_{m \ge 0} H^0(X, L^{m}))$
Let $X$ be a quasicompact and quasiseparated scheme and $L$ a semiample line bundle, i.e. the base locus $B_L:= \{x \in X \ \vert \ s(x)=0 \ \forall s \in \Gamma(X,L^{\otimes n})\}$ for $L$ is empty.
...
- 4,142
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Motivation for the functor of points description of $\mathbb{P}^n$
I'm teaching students motivations in scheme theory. It's known that a scheme $X$ is determined by the functor $Rings\to Sets, A\mapsto X(A)$. Also we know the scheme $\mathbb{P}^n=\mathbb{P}^n_{\...
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GIT quotients & Kähler differential forms
Let $X $ $\subset \Bbb P^n_k$ a smooth projective scheme (base field $k$ alg closed ) with projective linear action by a cyclic finite group $G =\Bbb Z/(m)$.
By "linear projectiveness" of ...
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191
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Mumford's conjecture on rational quotient
Let $S$ be smooth, projective variety over a characteristic zero field $k$ such that a general rational curve $C \subset S$ on it has semipositive normal bundle $O_C(C)$
(about terminology "...
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