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.
63 questions from the last 365 days
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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 ...
- 385
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57
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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 $E_j$ ...
- 328
5
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4
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1k
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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 ...
- 619
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69
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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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14
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1
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557
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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 ...
2
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1
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268
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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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1
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0
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117
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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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2
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0
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105
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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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112
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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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3
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1
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190
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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 ...
- 328
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0
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99
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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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110
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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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91
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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 $W(k)$ ...
- 519
1
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0
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219
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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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1
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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 ...
- 613
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0
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127
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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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1
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0
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161
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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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0
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123
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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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5
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1
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567
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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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4
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1
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252
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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} \...
- 318
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153
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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 ...
- 291
4
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0
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178
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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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1
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0
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177
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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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4
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1
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289
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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 $\operatorname{Proj}A/I$. ...
- 391
1
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1
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210
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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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4
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1
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236
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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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1
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142
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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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1
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103
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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 $p$. ...
- 321
1
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0
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71
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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 ...
- 321
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171
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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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1
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589
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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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2
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0
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69
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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 ...
7
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0
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274
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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.
...
5
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0
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166
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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 (...
- 413
5
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1
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350
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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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0
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118
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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 ...
- 6,018
3
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0
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229
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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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4
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198
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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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1
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1
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106
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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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4
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0
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135
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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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2
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169
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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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5
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1
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286
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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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1
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1
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364
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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 ...
- 6,018
2
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0
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233
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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 ...
- 91
2
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1
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178
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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 ...
2
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1
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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 ...
- 328
6
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1
answer
269
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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 ...
2
votes
1
answer
200
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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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1
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0
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52
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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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2
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0
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139
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What are the categories of IND and PRO schemes?
below is a mathexchange question with no answers so I drop it here.
I have some difficulties to figure out what the category of IND-schemes and PRO-schemes are, in particualer the relations with ...
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