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.
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Zariski Connectedness Theorem: From Analytic & Topological Viewpoint
Let $p:Y \to X$ be a proper, surjective map between smooth connected complex varieties $Y,X$, esp. $X$ unibranch. Assume that there exist a Zariski open (esp. dense) $U \subset X$ over which every ...
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L.c.i locus of Hilbert scheme of points on singular varieties
Let $X$ be an algebraic variety over $\mathbb{C}$. What can we say about the l.c.i. locus of $\text{Hilb}^n(X)$?
When $X$ is smooth, it is well-known that the l.c.i. locus of $\text{Hilb}^n(X)$ is ...
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Discrepancy of general element of linear system
Let $X$ be a normal scheme and $|D|$ a linear system on $X$.
In "Singularity of Minimal Model Program" by Janos kollar p249, it says,
If $X$ is a variety over $\mathbb{C}$, and $E_j$ ...
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Stable points in GIT: geometric picture
Is there a geometric picture justifying why "stable points" in GIT (Geometric Invariant Theory) are actually called "stable"? Stable, with respect to which effect? (Here, I have ...
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Descent of $G$-invariant formal system of parameters using GAGF
Let $R=(R,\mathfrak{m})$ be a comm local regular ring of char $\neq 2$ (ie $2 \neq 0$ in $R$) with maximal ideal $\mathfrak{m}$ of (Krull) dimension $2$, ie $R$ admits system of parameters $x,y \in \...
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What is the "schematic" point of view for regular polyhedra?
Last week, I read Wikipedia's article on Alexander Grothendieck. It lists his twelve greatest contributions to mathematics as accounted for in Grothendieck's own Récoltes et Semailles. The final item ...
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Jacobian fibration of elliptic fibration: basic relations between Enriques invariants
Let $f: X \to B$ be an elliptic fibration, so proper map from smooth surface $X$ onto smooth conn. curve over alg closed base field $k$ with connected fibers such that almost all fibers are elliptic ...
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Quotient of K3 surface: complex vs positive characteristic
Let $f: X \to X$ be a non-symplectic automorphism of finite order of complex projective K3 surface $X$. (Recall: Non-symplectic means that the induced action on $H(X,K_X)=H^0(X, \Omega_X^2)$ is not ...
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Torsion Freeness of Sheaf of Kähler Differentials
Let $X$ be an irreducible scheme over some base field $k$. Consider the sheaf of Kähler differentials $\Omega_{X/k}$. Let $w: \Omega_{X/k} \to j_* \Omega_{K(X)/k}$ be natural map induced by enbedding ...
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Irregularity of surfaces for dominant maps
I have a question about an argument in the proof of Lemma 1.2.(1) in Quotients of K3 surfaces modulo involutions by D. Q. Zhang:
Let Let $(X, \sigma)$ be X be a smooth projective K3 surface with an ...
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Irreducibility under etale ring map
Let $A\rightarrow B$ be a etale ring map between finite type algebra over algebraically closed field $k$.
If $A$ is one dimensional integral domain, is $B$ direct product of finite type integral ...
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Quotients of K3 surfaces vs cyclic covers
Let $X$ be an algebraic K3 surface (for sake of simplicity, with base field of char $\neq 2$) and $f: X \to X$ a non-symplectic morphism (i.e. non-symplectic in sense of that that the induced action ...
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Lefschetz Theorem in Dolgachev's On automorphisms of Enriques Surfaces
Let $F$ be a Enriques surface over $\Bbb C$. I have a question about a detail in the proof of Proposition 2.1. from Dolgachev's On automorphisms of Enriques surfaces.
This 2.1. Proposition. states ...
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Lifting smooth proper varieties over finite fields to finite extensions of $W(k)[1/p]$
Let $k$ be a finite field of characteristic $p > 0$, and let $X$ be a smooth proper variety over $k$. It is generally unknown whether $X$ admits a smooth proper lifting over $W(k$, where $W(k)$ ...
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Quotient of K3 surfaces by non-symplectic automorphism of finite order
Let $X$ be a $K3$ surface and $f: X \to X$ a non-symplectic morphism (ie non symplectic in sense of that that the induced action on $H(X,K_X=H^0(X, \Omega_X^2)$ is not trivial) of finite order.
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Relation between quot scheme of birational curve
I am very new to algebraic geometry. Currently reading about Hilbert and quot scheme. I want to know more about the structure and properties of Hilbert and quot schemes over curves. My question is the ...
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Relative minimal models of pencils of surfaces
I would like to ask for recomendation for literature on theory relative minimal models of surfaces, where "relative" in sense of that the study objects are not surfaces alone (="absolue ...
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Special elliptic pencil of an Enriques surface (arguments in a proof)
I have a couple of questions about arguments in the proof of Lemma 2.6 (see absol page 199, rel p 9) from Shigeyuki Kondo's paper Enriques surfaces with finite automorphism groups:
The setup: Let $Y$ ...
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Counit map surjective
Let $X \to Y$ a (set theoretically) surjective morphism of schemes, $L$ a line bundle/invertible sheaf on $X$ (maybe more generally a locally free coh sheaf, but let's stick firstly on invertible ...
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Dualizing sheaf of nodal curve
Let $C$ be a connected, nodal (I'm working with definition from Alper's notes on Stacks & Moduli, see p 210), projective curve over an alg closed field $k$, beeing everywhere smooth except at a ...
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Multiplicative cancellation for trivial vector bundles
Let $X$ be a scheme, ${\mathscr L}$ an invertible ${\mathscr O}_X$-module, and $d$ a positive integer. If ${\mathscr L}^{\oplus d} \simeq {\mathscr O}_X^{\oplus d}$, does it follow that ${\mathscr L} \...
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A schematic representability of an algebraic space with group action
In the book "Néron Models" (BLR), there is a statement as follows (on page 164):
Let $S$ be a locally noetherian scheme and let $G$ be a smooth algebraic group space over $S$ with connected ...
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Splitting of counit-trace map for $\ell$-adic sheaf $\Bbb Q_{\ell}$
I have a question about following argument on page 3 in paper arxiv.org/abs/1702.04404 by Will Sawin (Proposition 3):
The claim is that for a finite, dominant map $f:X \to Y$ between varieties $X,Y$ ...
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Prop 1.3 in "Birational geometry of algebraic varieties": specialization of rational curves
I have a couple of questions about some arguments in proof of Proposition 1.3 from Birational geometry of algebraic varieties by Kollár and Mori (p 8):
Proposition 1.3. [Abh56, Prop. 4] Let $X$ be ...
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Counter example to every closed subscheme $\operatorname{Proj} A$ is of the form $\operatorname{Proj}A/I$
I was under the impression that for a positively graded ring $A$ (not necessarily generated in degree $1$) that every closed subscheme of $\operatorname{Proj}A$ was of the $\operatorname{Proj}A/I$. ...
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Characterize descents of geometric finite étale cover by means of homotopy exact sequence
Let $X/k$ be a geometrically connected $k$-variety (=separated of finite type, esp quasi-compact; the base field $k$ assumed to be separable, so $\overline{k}=k^{\text{sep}}$), $\overline{X} := X \...
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Ampleness verifiable over faithfully flat cover
Let $X$ be a Noetherian scheme over a field $k$ and $\mathcal{L}$ an invertible sheaf. Recall $\mathcal{L}$ is called ample iff for every coherent $\mathcal{M}$ there exist a $n_0(M)$ such that for ...
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$G$- Fixed Point Scheme explicitly
Let $G$ be an abstract finite group acting on a separated $k$-scheme $X$. ($k$ a field; note we can canonically promote $G$ to a $k$- scheme). Then a result by Demazure and Grothendieck (in "...
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Calculation of intersection multiplicity after the restricting to a fiber
Let $X\to\operatorname{Spec} \mathbb Z$ be an arithmetic surface which is projective, regular and integral. Let $D$ and $E$ two divisors intersecting at a point $x\in X$ that lies over the prime $p$. ...
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Explicit field extension for semistable models of curves
The paper arxiv:1211.4624 briefly summarizes the way to find a semistable model of a curve $X/K$ (the existence of the model is ensured by the Deligne-Mumford theorem). Specifically the author says ...
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Nice blowups are universal algebraic fiber spaces?
We say that a proper (maybe projective) morphism $f:X \to Y$ is a universal algebraic fiber space if $f_* O_X = O_Y$ holds universally. (This means: for any morphism $Y' \to Y$, if $X' = Y' \times_Y X$...
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Argument for non-existence of elliptic curve over $\mathbb{C}[t, t^{-1}]$
I have a couple of questions about this answer by Noam D. Elkies showing that there exist no elliptic curve $E$ over $\mathbb{C}[t, t^{-1}]$ having nonconstant $j_E$-invariant.
The strategy is to ...
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Irreducibility of Białynicki-Birula cells
Let $X\subset \mathbb{P}^n$ be a smooth complex projective variety, and consider a non-trivial action of $\mathbb{C}^*$ on $X$. For any connected fixed component $Y$ of the fixed point locus, we may ...
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Is every normalization a blowup?
I asked this at math.stackexchange, but received no reply.
Is the normalization of a variety always a blowup along some coherent ideal sheaf? If not, I would like to see a concrete counter-example.
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Galois ascent - When is a variety a Weil restriction?
Let $L|K$ be a finite Galois extension of degree $d$ and $X$ be a variety over $K$.
Is there a simple criterion, similar to Galois descent, allowing to determine whether $X$ is the Weil restriction (...
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Dévissage of stratified structures in Grothendieck's "Esquisse d’un programme"
I have a question about the intuition behind Grothendieck's proposed notion of so called "Tame topology" in his Esquisse d’un programme. Grothendieck insisted that theory should admit “...
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Induced action on infinitesimal thickenings by an algebraic group
Let $X$ be an irreducible locally noetherian $k$-scheme (for $k$ any field), $G$ an algebraic group acting on $X$ via $a:G \times X \to X$ and $x \in X$ a closed point, which is by Zariski's lemma ...
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Action of an algebraic group $G$ on a scheme $X$ with fixed rational point
Let $X$ be an irreducible locally noetherian $k$-scheme ($k$ any field) and $G$ an algebraic $k$-group acting on $X$.
Proposition 3.1.6 in these notes by M. Brion claims
Let $a : G \times X \to X$ be ...
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Questions about the fixed point functor $X^G$ of a $G$-scheme
Let $X$ a (locally Noetherian; but not sure if that's really matter) $k$-scheme, $G$ a $k$-group scheme acting on $X$ via morphism $a:X \times G \to X$.
The fixed point functor of $X$ (where $X$ is ...
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Torsor of finite presentation and surjectivity of map of $\overline{k}$-valued points
I have a question about the content of remark 2.6.6. (i) (p 18) from M. Brion's notes on structure of algebraic groups.
Let $G$ be a group scheme over certain fixed base field $k$ (as all other ...
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Specialization map Chow groups preserves algebraic equivalence
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 ...
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Principle of degeneration as precursor of Zariski's connectedness theorem (geometric intuition)
I have following question about so-called "principle of degeneration"
in algebraic geometry (which in modern terms is an immediate consequence
of Zariski's main theorem and goes in it's ...
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Origin of the name Trace resp Integral symbol for the trace map of Dualizing Sheaf
Let $X \subset \mathbb{P}^n_k$ be a normal projective subscheme over $k$ of dimension $n$. The dualizing sheaf is in context of Serre duality a pair $(\omega_X,t)$ (which exists in that case) ...
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Proj construction and nilpotent homogenous elements in graded ring
Let $A= \bigoplus_{n \ge 0} A_n$ be a commutative Noetherian graded ring and $f \in A_d$ a nonzero homogeneous element of degree $d>0$. The natural ring map $q:A \to A/(f)$ induces a well defined ...
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Representability of moduli problem of elliptic curves with complex multiplication
I'd like to know whether the moduli problem for elliptic curves with complex multiplication by a fixed imaginary quadratic number field $K$ (and with suitable level structure to be picked) is ...
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Blowup formula for a morphism
Let $f: X\to S$ be a smooth projective morphism between smooth schemes over $\mathbb C$, $i: Z \to X$ a closed subscheme of codimension $c$, also smooth over $S$, and let $g: Y\to S$ be the blowup ...
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210
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Surjective étale map from simply connected curve over $\mathbb{C}$
Let $X$ be a simply connected algebraic curve over $\mathbb{C}$ and $f:X\rightarrow \mathbb{A}^1_{\mathbb{C}}$ is a surjective étale map.
Then is it true $f$ is finite?
All the domains of non finite ...
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Criteria for when Gauss-Manin sheaves are vector bundles
Let $(X,D_X)$ and $(S, D_S)$ be smooth normal crossings pairs over $\mathbb C$; i.e. smooth schemes of finite type over $\mathbb C$ with a normal crossings divisor. If $f:X \to S$ is a proper, flat ...
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Find stratification to decompose constructible sheaf to constant parts (example from Wikipedia)
I have a question about techniques used in determining the stratification over which a constructible sheaf falls into even constant pieces demonstrated on this example from Wikipedia.
Let $f:X = \text{...
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Symmetric 0-dimensional schemes with generic Hilbert function and Grassmannians
I've came across this problem while thinking about some properties of fat schemes.
Let me give you an explicit (motivating) example:
We have $S=\mathbb{C}[x,y,z]$, the coordinate ring of $\mathbb{P}^2$...
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