All Questions
486 questions with no upvoted or accepted answers
1
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104
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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
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 $\mathbb{P}^2$...
- 1,343
1
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0
answers
218
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Interpretation of model theory in algebraic geometry
I found a paper Some applications of a model theoretic fact to (semi-) algebraic geometry by Lou van den Dries. In this paper, the author uses model theoretical methods to prove the completeness of ...
- 328
1
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0
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169
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Étaleness of Isom scheme $\operatorname{Isom}_S(X,Y)$
Let $S$ be a quasi-projective scheme over base field $k$ and $X, Y$ two finite étale schemes over $S$ and assume we are in situation we know that the isom space $\operatorname{Isom}_S(X,Y)$ exists as ...
- 6,006
1
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0
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137
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Locus where a family of cycles is rationally trivial is closed?
Let $B$ be a smooth quasi-projective variety over a field of characteristic zero.
Let $\pi\colon \mathcal{X} \rightarrow B$ be a smooth and projective morphism with geometrically integral fibres. Let $...
- 984
1
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0
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98
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Adjunction correspondence for Blow up of double point
Let $C$ a curve over an algebr closed field $k$ with a singular double point singularity at $x$ and $\pi: C' \to C$ the blowup in $x$ and let $x_1,x_2 \in C'$ be the two points over $x$.
Why holds for ...
- 6,006
1
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0
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50
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Do parabolic/Levi pairs admit dynamic descriptions over disconnected base?
In Gille, Thm. 7.3.1, it is proven that given a reductive group scheme $G \to S$ over a connected base $S$, every parabolic-Levi pair $(P, L)$ over $S$ admits a dynamic description, i.e. is of the ...
- 605
1
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0
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264
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An algebraic stack is an algebraic space if and only if it has the trivial stabilizer group
Let $G\to S$ be a smooth affine group scheme over a scheme. Let $U$ be a scheme over $S$ with an action of $G$. Let $[U/G]$ be the quotient stack.
In Alper's note: Stacks and Moduli, there is a result ...
- 11
1
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0
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75
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Idempotent completeness
We say a category $\mathcal{N}$ is exact if it is additive and is endowed with an exact structure. In brief, it is an additive category with a predetermined class of short exact sequences in its ...
- 167
1
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0
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210
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Definition of “morphism of schemes that induces a bijection between irreducible components ”
$\def\sO{\mathcal{O}}\def\sF{\mathcal{F}}$On the Stacks Project there are several instances where the seemingly undefined notion of a “morphism of schemes that induces a bijection between irreducible ...
1
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0
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79
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Is every classical prevariety the set of $k$-rational points of an schematic prevariety? (when $k$ is not algebraically closed)
$\def\cpvar{\mathsf{CPVar}}
\def\spvar{\mathsf{SPVar}}
\def\Spec{\operatorname{Spec}}
\def\class{\mathrm{class}}
\def\sO{\mathcal{O}}
\def\Hom{\operatorname{Hom}}$This question is a follow-up to this ...
1
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0
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119
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Action by finite abstract group on affine scheme
Let $X:=\operatorname{Spec}(R)$ an affine Noetherian scheme and $G$ a finite group acting on $X$. Then it is known that the quotient $Y=X/G$ exists as affine scheme $\operatorname{Spec}(R^G)$, let set ...
- 6,006
1
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0
answers
88
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Bad primes of twists of modular curves $X_E^{-1}(p)$
I am interested in a good reference for reading about primes of bad reduction for the modular curve $X_E^{-}(N)$, which is a twist of $X(N)$ parametrizing all elliptic curves $E'$ whose $N$-torsion ...
- 637
1
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0
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137
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Pushforward of locally free sheaf by open immersion
Say $X$ is a smooth variety (even just $\mathbb{A}^n$) and $j\colon U\hookrightarrow X$ is an open immersion with $X - U$ of codimension 2 such that $E$ is a locally free sheaf on $U$. Since $X$ is ...
- 43
1
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0
answers
150
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Relative compactification without resolutions of singularities
Let $Y$ be a smooth proper variety over a field $k$, let $X$ be a smooth variety over $k$, $U\hookrightarrow X$ the complement of a strict normal crossing divisor and $\phi\colon U\to Y$ a map. By ...
- 309
1
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0
answers
29
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Connected components of Isotropy types as strata of Poisson leaves
Let $X$ be a smooth affine variety with an algebraic symplectic form $\omega$. Let $G$ be a finite subgroup of the group of symplectomorphisms of $X$.
We can say that $X$ is trivially a normal variety ...
- 609
1
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0
answers
120
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Linear span of tangential variety
Let $X \subset \mathbb{P}^N$ be a projective variety of dimension $n$. Let us denote with $TX=\bigcup_{x \in X}\mathbb{T}_xX$ the tangential variety, where $\mathbb{T}_x X$ is the projective tangent ...
- 1,343
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88
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Invariance of numerical class of a curve in Higgs-Grassmann schemes
Premise
Let $X$ be a projective variety of dimension $n\geq1$ over an algebraically closed field of characteristic $0$.
A Higgs sheaf $\mathfrak{E}$ is a pair $(E,\varphi)$ where $E$ is a $\mathcal{...
- 828
1
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0
answers
44
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What is the kernel of the differential of the orbit-stabilizer map for nonsmooth stabilizers?
$\newcommand{\Lie}{\operatorname{Lie}}$Let $G$ be a smooth linear algebraic variety over perfect field $k$, acting on a separated variety $X$, and for $x \in X(k)$ write $G_x$ for the scheme-theoretic ...
- 605
1
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0
answers
106
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Joins of (closed) subschemes and Zariski-local Z-functors
$\newcommand\Aff{\mathrm{Aff}}\newcommand\cRing{\mathrm{cRing}}\newcommand\Sch{\mathrm{Sch}}$Equip $\Aff = \cRing^\text{op}$ with the Zariski Grothendieck-topology. There are nested categories:
$$\Aff\...
- 775
1
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0
answers
205
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Projective scheme over the integers
Let $X$ be a projective scheme over $Spec(\mathbb{Z})$. Let $X_{p}$ be the reduction at $p$ of $X$. If for any prime $p$, $X_{p}$ is normal, can we deduce $X$ is normal? Or any counterexamples?
- 653
1
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0
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114
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Check whether a closed point of a Noetherian affine scheme is a local complete intersection
Suppose that $k$ is an algebraically closed field and $A$ is the ring $k[a,b,c,d]/(ac-b^2,bd-c^2,ad-bc)$. Let $X$ be Spec$A$, and $m$ be the maximal ideal of $A$ generated by the quotient images of $a,...
- 639
1
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0
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122
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How to compute the G-theory groups of a blow-up of Noetherian schemes
Suppose that $k$ is an algebraically closed field and $R$ is a finitely generated $k$-algebra such that if $X$ denotes Spec$R$, then the only closed, singular point of $X$ is the origin. Let $\tilde{X}...
- 639
1
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0
answers
335
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Meaning of "cut out (scheme-theoretically)"
Let $V$ be a projectively normal closed subvariety of some projective space over an algebraically closed field $\mathbb{K}$. Let $R$ be the local ring at the vertex of the affine cone over $V$ ($R$ is ...
- 839
1
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0
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324
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On construction of Hilbert and Quot schemes
I have some questions regarding the strategy of the proof of the existence of Hilbert and Quot schemes (I will focus on the latter since it's more general), as in the book Fundamental Algebraic ...
- 1,906
1
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0
answers
186
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Zeta functions of schemes of finite type over $\mathbb{Z}$
Let $X$ be a scheme of finite type over $\mathbb{Z}$. In Section 11 of my 2008 paper in J. Number Theory, "Ring structures on groups of arithmetic functions," I define an additive analogue $...
- 4,985
1
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answers
156
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integral subschems of geometrically integeral schemes
Let $X$ be a geometrically integral surface over a field $k$ and let $C$ be an integral curve on $X$. I want to know whether $C$ is geometrically irreducible over $k$ or not.
- 19
1
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0
answers
124
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Combinatorics of projective planes over commutative rings
An axiomatic projective plane is a point-line incidence structure with the following axioms:
any two distinct points are collinear (via a unique line);
any two distinct lines meet in a unique point;
...
- 4,605
1
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0
answers
105
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Interesting property of a divisor contained in special fiber
Let $(A, \mathfrak{m}, \kappa=A/\mathfrak{m})$ be a local ring and $f:X \to \operatorname{Spec} (A)$
a scheme. Let $D \subset X$ a divisor on $X$ contained in special fiber $D \subset f^{-1}(\sigma_{\...
- 6,006
1
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0
answers
96
views
Valuation diagram of scheme
Let $f: X\rightarrow Y$ be a morphism of schemes, then if $f$ is quasi-compact, then there exist a valuation ring $A$ and its fraction field $K$ satisfying the following commutative diagram. The proof ...
user169802
1
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0
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71
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Gluing categorical limit over subgraphs
Let $C$ be a category, and $\Gamma$ a graph in $C$. Under good conditions it makes sense to talk about the limit $\lim \Gamma$ of $\Gamma$ in $C$.
Suppose $\Gamma$ is the union of two subgraphs $\...
- 5,230
1
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0
answers
189
views
Existence of integral extension of DVR satisfying some conditions
Let $X^{\prime}$ and $X$ be integral noether schemes over $\mathbb{C}$, and $p:X^{\prime}\rightarrow X$ be a surjective morphism.
Let $R$ be any discrete valuation ring over $\mathbb{C}$ with its ...
- 297
1
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0
answers
51
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Do we have $\cap_{P\in S}\left(\mathcal{O}_P+(1-\tau)(K)\right)=\left(\cap_{P\in S}\mathcal{O}_P\right)+(1-\tau)(K)$?
Let $\mathbb{F}$ be a finite field and let $q$ be its number of elements. Let $(C,\mathcal{O}_C)$ be a geometrically irreducible smooth projective curve over $\mathbb{F}$ and let $K=\mathbb{F}(C)$ be ...
- 1,479
1
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0
answers
135
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Projection from closure of locally closed subscheme is Etale
Let $S$ an arbitrary scheme and denote by $\Delta: S \to S \times S$ the diagonal immersion and $p_i: S \times S \to S$ the both projections to first resp second factor. (in following we will wlog ...
- 6,006
1
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0
answers
188
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Affine scheme as algebraic space
We working in the following with Knutson's definition of an algebraic space
(ie via equivalence relation; there is also another equivalent def via
sheaves but let us work here with the following one):
...
- 6,006
1
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0
answers
212
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Dimension of a linear system of divisors on singular curve
Consider an singular irreducible plane curve $C \subset \mathbb{P}^2_k$ of degree $d>1$ over algebraically closed field $k$ which is given as vanishing locus $C=V(f(x,y,z))$ of a $f \in k[x,y,z]$ ...
- 6,006
1
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0
answers
165
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Discriminant ideal in a member of Barsotti-Tate Group
Let $S = \operatorname{Spec} R$ an affine scheme (in our case latter a complete dvr) and $p$ a prime. Then Barsotti-Tate group or $p$-divisible group $G$ of height $h$ over $S$
is an inductive system
...
- 6,006
1
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0
answers
56
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local acyclicity when restricting to an hypersurface
Let $X$ be a smooth scheme over $\mathbb{C}$ and a constructible sheaf $K$ of complex vector spaces on $X\times\mathbb{A}^1$ and a function $g:X\rightarrow \mathbb{A}^1$.
Suppose that $K$ is locally ...
- 3,472
1
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0
answers
410
views
Separable morphism of curves
A proof from Janos Kollar's Lectures on Resolution of Singularities Kollar (p 37) works as follows:
Theorem 1.58 (M. Noether, 1871). Let $k$ be an algebraically closed
field and $C \subset \...
- 6,006
1
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0
answers
299
views
Application of Galois descent
I not understand an assumption done at the beginning of the proof of Rigidity lemma in Moonens and van der Geers book about Abelian variaties (page 12). Here is it:
Question: Why the assumption $k= \...
- 6,006
1
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0
answers
275
views
Pullback of ideal sheaf under base change by completion of base ring
Assume $R$ is a discrete valuation ring with uniformizing parameter $t$, i.e. $\mathfrak{m}_R=(t)$. We denote $\widehat{R}$ the completion of $R$ with respect to $(t)$. Let $Y$ be a flat locally ...
- 6,006
1
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0
answers
149
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Formula for fibre square (from Fulton's Intersection Theory)
I have a question about argumentation used in the proof of Proposition 1.7 in Fulton's book on Intersection Theory on page 18:
Proposition 1.7 Let
$\require{AMScd}$
\begin{CD}
X' @>{g'}>> ...
- 6,006
1
vote
0
answers
246
views
Frobenius twist of a field
Let $k$ be a field of characteristic $p>0$ (not necessarily perfect). Consider the Frobenius endomorphism $F : k \to k$, $x \mapsto x^p$. I am curious about what happens when we take $k$ as a $k$-...
1
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0
answers
157
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Application of Stein factorisation: rigidity lemma
Let $X,Y$ Noetherian schemes and $f:X \to Y$ proper map. The Stein factorisation factorizes $f$ as $X \xrightarrow{g} Spec \text{ } f_* \mathcal{O}_X \xrightarrow{h} Y$ with $h$ finite and $g$ has ...
- 6,006
1
vote
0
answers
146
views
Factoriality of schubert cells in affine flag variety
Take for simplicity $G=SL_n$ and consider the affine flag variety $Fl=G(\mathbb{C}((t)))/I$ for $I$ the Iwahori corresponding to the Borel of upper triangular matrices of determinant one.
For each $...
- 3,472
1
vote
0
answers
92
views
Topological space modeled by special topological structures
Let $X$ be a topological space. Suppose it is "modeled by" topological spaces of the form $\text{Spec}(A)$ for some commutative ring $A$, then, (along with some other conditions/structure), we call $...
- 6,023
1
vote
0
answers
95
views
Units in the coordinate ring on a reductive group
Let $K$ be a field and $G$ a connected reductive group over $K$.
Can we describe $K[G]^{*}$?
- 3,472
1
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0
answers
189
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Relate Codimensions of Integral Schemes and their Generic Fibers
I have a question about an argument in the proof of Thm 8.2.5 (Dimension formula) from Liu's "Algebraic Geometry" (page 334):
Firstly the problem is reduced to the affine case with $Y:=Spec \text{ }\...
- 6,006
1
vote
0
answers
160
views
Riemann–Hurwitz Formula for Normal Projective Curves
My question refers to the proof of Theorem 7.4.16 from Liu's "Algebraic Geometry" at page 290 :
QUESTION: Why does $e'_x= length_{\hat{B}}(W_{\hat{B}/\hat{A}}/\hat{B})$ hold?
During the proof we ...
- 6,006