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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.

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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 $C^\infty(M, \mathbb{R})$. ...
Leandro Lorenzetti's user avatar
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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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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 ...
user267839's user avatar
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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 $...
user267839's user avatar
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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 ...
Anubhab Pahari's user avatar
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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) ...
Manoel's user avatar
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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 ...
Alchemist's user avatar
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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 $...
PauotCC's user avatar
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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. ...
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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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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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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 "...
user267839's user avatar
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1 answer
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Let $X$ be a scheme over some base field $k$ and $L$ a line bundle. Consider the induced rational maps $\varphi_{|L^m|} : X \dashrightarrow \mathbb{P}(H^0(X, L^m)^*)$ induced by global sections. It is ...
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Let $R$ be a discrete valuation ring. Let X be a smooth proper scheme over $R$. Let $\eta\in \mathrm{Spec} R$ be the generic point. Then by 21.6.11 of EGA IV4, the restriction $\mathrm{Pic}(X)\to \...
Doug Liu's user avatar
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Let X be a singular projective variety over $\mathbb{F}_p$. Then for $i\ge0$, the crystalline cohomology group $H^i_{\mathrm{cris}}(X/\mathbb{Z}_p)[1/p]$ is a $\mathbb{Q}_p$-vector space. Is it of ...
Doug Liu's user avatar
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