All Questions
22,546 questions
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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 ...
- 6,048
3
votes
3
answers
1k
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The non-existence of the fine moduli scheme of vector bundles. Why?
The reference I am using is Hoffmann - The moduli stack of vector bundles on a curve. The question is about the moduli space of vector bundles. I am trying to understand why the fine moduli scheme ...
- 3,439
4
votes
3
answers
266
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References for $K$-orbits in $G/B$
Let $G$ be a reductive group, $K$ a symmetric subgroup of $G$ (e.g., fixed point of an involution), and $B$ a Borel subgroup of $G$. Then it is well known that $G/B$ has finitely many $K$-orbits. ...
- 1
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0
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190
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About Chern classes via Atiyah class
I am trying to understand a construction of the Chern classes of a vector bundle $\mathcal{E}$ via the Atiyah class, like is done in this text and here in section 1.4. I am interested in the case ...
- 4,713
2
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0
answers
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 ...
- 6,048
41
votes
1
answer
2k
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Implications and consequences of the recent proof of the geometric Langlands conjecture
I am a beginner in mathematical physics and geometric Langlands, having very limited knowledge in both fields so far.
The proof of geometric Langlands conjecture is published a few months ago. What ...
- 24.1k
2
votes
0
answers
127
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Nonabelian Hodge correspondence for $\mathbb{G}_m$
Please excuse me if this question is too naive. I know very little about the nonabelian Hodge correspondence but I am trying to understand how the correspondence works in the simplest case of the ...
- 3,446
12
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0
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605
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What's a holonomic D-module from the point of view of de Rham spaces?
Let $X$ be a smooth algebraic variety over $\mathbb{C}$. We can consider its de Rham space $X_\text{dR}$ as the sheaf on $(\textsf{Sch}/\mathbb{C})_\text{ét}$ defined by $X_\text{dR}(S):= X(S_\text{...
- 771
1
vote
1
answer
207
views
Is the vector bundle over a vector bundle, a vector bundle over the base scheme?
Suppose we have $E\to X$, a vector bundle over $X$ and $E'\to E$ a vector bundle over $E$. Composing the structure maps gives a smooth scheme $E'\to X$ over $X$. My question is, when is this a vector ...
- 7,377
4
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1
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176
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Grothendieck construction on fibred categories/stacks
This question is related to a previous question of mine, which has so far gone unanswered.
For a fixed site $\mathcal C$, the fibred categories over $\mathcal C$ form a (strict) $2$-category, see here....
- 15.9k
0
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0
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98
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Does the smooth locus of any toric variety built from a fan always contain a rational point?
Let $k$ be an arbitrary field and $X$ be a toric variety built from a fan, defined over $k$.
Does the smooth locus of $X$ always contain a $k$-rational point? Why?
- 639
4
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1
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172
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How to exchange left and right $\mathcal{D}_X$-modules when $X$ is not smooth?
When $X$ is a smooth scheme (over something of characteristic $0$), one can exchange left and right $\mathcal{D}_X$-modules ($\mathcal{D}_X$ means the sheaf of differential operators) by tensor with (...
- 2,368
3
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246
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Fundamental group of degree 4 del Pezzo surface minus 16 (-1)-curves [Reference request]
Let $S$ be a degree $4$ del Pezzo surface (over $\mathbb{C}$).
That is, $5$ points blow-up of $\mathbb{P}^2$, or $4$ points blow-up of $\mathbb{P}^1 \times \mathbb{P}^1$.
The classical fact is that $...
- 111
1
vote
2
answers
341
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Small contraction for Hyperkähler Varieties
I have the following basic question. Everything is over $\mathbb{C}$.
Let $X$ be a hyperkähler (irreducible holomorphic symplectic) variety and we consider a small contraction $f\colon X \rightarrow ...
- 115
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0
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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 ...
- 6,048
3
votes
0
answers
125
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Parametrization of indecomposable modules via quiver varieties
Let $k$ be an algebraically closed field, $Q$ a quiver without oriented cycles and $m^\alpha (Q)$ the variety of quiver representations with dimension vector $\alpha$. There is a canonical algebraic ...
- 696
1
vote
1
answer
224
views
Proper smooth pushforward of vector bundle is a vector bundle?
Suppose $X$ and $Y$ are algebraic varieties over a field $k$, and $f:X
\to Y$ is proper smooth. Then for a vector bundle $E$, and any $i\ge 0$, do we have $R^if_*(E)$ locally free? I know we need the ...
- 16.2k
1
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0
answers
113
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Analytic vector bundle from an etale local system is algebraic?
Suppose $X$ is an algebraic variety over $\mathbb C$, and $\mathbb L$ is a $\mathbb Q_p$-local system on $X_{et}$, then it corresponds to a representation $\pi_1(X_{et})\to GL_n(\mathbb L)$. Since ...
- 785
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9
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9k
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How to motivate and present epsilon-delta proofs to undergraduates?
This would seem to be a common question, but I am surprised not to see it already asked and answered on MO!
I am teaching an undergraduate course, and I want to teach them to construct basic epsilon-...
CommunityBot
- 1
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votes
1
answer
128
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Chern Classes of $\mathcal{O}_E(1)$ on $\mathbb{P}(E)$ for $E = \mathcal{O} \oplus \mathcal{O}(n) \to \mathbb{P}^2$
Let $E =\mathcal{O} \oplus \mathcal{O}(n) \to \mathbb{P}^2$ and denote by $\mathcal{O}_E(1)$ the dual of the tautological bundle.
How can I compute $c_1^2(\mathcal{O}_E(1)), c_1^3(\mathcal{O}_E(1))$, $...
- 39.3k
5
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1
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290
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Why does a line bundle on an abelian variety give a group extension only if it is algebraically trivial?
If $X$ is an abelian variety and $L$ is a line bundle, deleting the zero-section one obtains a diagram
$$
0 \to \mathbb G_m \to Y \stackrel{\phi}{\to} X \to 0
$$
where $\mathbb G_m = \phi^{-1}(0)$,
...
CommunityBot
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2
votes
0
answers
95
views
Pullback of an ample bundle under an embedding is ample
In Example 11.8 on JP Demailly's book on Complex Analytic and Differential Geometry it is being said that
The pullback of a (very) ample line bundle by an embedding is clearly also (very) ample.
I ...
- 21
1
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0
answers
97
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Weil restriction of cycles and norm algebra
This question is on a concrete descrption of weil restricton of an affine algebra.
Let L/K be a Galois extension. Since I only care about the quadratic case, we may assume that $\Gamma:=\operatorname{...
- 63.9k
1
vote
0
answers
102
views
On descending a section of a morphism between schemes from formal completion to étale local
Here's the case, which arises from the context of doing infinitesimal deformation. Given a DVR $(S,\mathfrak{m},\kappa)$ we have the completion $\hat{S}$ with respect to $\mathfrak{m}$. Say we have a ...
- 12.9k
0
votes
0
answers
170
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Theorems related to Chevalley's theorem
Recently I have read Chevalley's theorem of a complete local ring which basically says that if $(R,\mathfrak{m})$ is a complete local ring and if $\{b_n\}$ be a sequence of ideals such that $b_n \...
- 63.9k
3
votes
1
answer
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 ...
- 10.7k
1
vote
0
answers
219
views
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.
...
- 6,048
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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 ...
- 63.9k
1
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70
views
Degree axiom for P1 or P2
I am getting stuck on equation (7.33) on p. 192 of Cox-Katz's Mirror Symmetry and Algebraic Geometry. This concerns the degree of a cohomology class used as input for a Gromov-Witten invariant.
Let $X$...
- 511
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votes
1
answer
131
views
Poset definition of dimension
Let $\mathsf{k}$ be an algebraically closed field and $X$ an abstract variety (an integral separated scheme of finite type over $\mathsf{k}$).
Is there any way to define the usual dimension of $X$ ...
- 3,318
5
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2
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218
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Smooth toric variety which is a cube is a bott tower (reference request)
According to Lee, Masuda and Park (page 3), the following result is "well-known in toric topology". I've found a proof, but I would like a published reference.
Let $X$ be a toric variety. ...
- 66
4
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1
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165
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Describing the compactified Jacobian of a nodal curve
$\DeclareMathOperator{\Pic}{Pic}$Let $C$ be an integral projective curve over $\mathbb C$, which is smooth except for a single node $x\in C$. Let $M$ be the moduli space of stable torsion-free sheaves ...
- 1,286
9
votes
2
answers
803
views
Explanation for Lurie's SAG Remark 25.1.3.7
I am trying to understand the theory of simplicial commutative rings or animated rings. I just find a remark in Lurie's book Spectral Algebraic Geometry:
Remark 25.3.1.7. Let $f : R[x_1,\ldots ,x_n]\...
- 56.9k
0
votes
0
answers
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 ...
- 6,048
3
votes
0
answers
164
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Pro-algebraic fundamental groups
Let $X$ be a smooth projective variety over an algebraically closed field $K$ of characteristic zero and fix a point $x\in X(K)$.
We can associate to $X$ two Tannakian categories: the category of ...
- 3,446
0
votes
1
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408
views
Rational normal curve as determinantal variety
We work here over complex numbers. Let $\Omega(Z)$
\begin{pmatrix}
L_1 & L_2 & ... & L_n \\
M_1 & M_2 & ... & M_n\\
\end{pmatrix}
be a $2 \times n$ matrix of homogeneous ...
CommunityBot
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2
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0
answers
92
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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
21
votes
2
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2k
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Naive question about constructing constructible sheaves.
In algebraic geometry, an etale sheaf on a Noetherian scheme is called constructible if the scheme has a finite stratification by locally closed subschemes such that the pullback of the sheaf to each ...
- 28.7k
0
votes
0
answers
61
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$\mathcal{R}$ is finite over $L_0[e,A]$
Let $\sigma:L \longrightarrow L$ an automorphism of infinite order, where $L_0 \subset L$ is its fixed field. Let $R$ be a commutative subring of $L\{\sigma\}$. Let
\begin{equation}
\mathcal{R}=\...
- 69
1
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0
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192
views
When are the complex points of a scheme an analytic manifold/space
Original Question: Let $X$ be a regular, projective, flat scheme over $\mathbb{Z}$. Let $X(\mathbb{C}$) be the set of complex points of $X$. Why is $X(\mathbb{C}$) a complex analytic manifold? I am ...
0
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0
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89
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Contraction of extremal ray on a smooth projective threefold
I have some issues about understanding the contraction of extremal ray in a concrete situation:
Let $\mathcal{E}=\mathcal{O}_{\mathbb{P}^1\times \mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1\times \...
- 41
4
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2
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305
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Is the property of being a torsor for a smooth affine group scheme detectable on the small étale site?
Let $S$ be a scheme, $G$ a smooth $S$-affine $S$-group scheme, and $X$ an $S$-scheme with a ($S$-linear) $G$-action $\alpha : G\times X\rightarrow X$.
Let $h_X$, $h_G$ be the representable sheaves on ...
- 921
1
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0
answers
84
views
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
0
votes
0
answers
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 ...
- 6,048
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votes
0
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44
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Sufficient conditions for a homogeneous polynomial to have a continuous right inverse
this is a question that continues a series of questions I'm coming up with on homogeneous polynomials, like for example this one.
For now I can prove that a homogeneous polynomial $f:\mathbb R^n\to \...
- 311
1
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0
answers
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 $Y$ ...
- 6,048
4
votes
1
answer
250
views
Galois action on the pro-algebraic completion of the singular fundamental group
Let $X$ be a smooth variety over a field $K \subset \mathbb{C}$. The singular fundamental group $\pi_1(X^{\text{an}}, x)$ generally does not carry an action of the absolute Galois group $\operatorname{...
- 12.9k
1
vote
1
answer
156
views
Determinant bundle over homogeneous varieties
I am looking for a way to compute the determinant of a homogeneous vector bundle over any homogeous variety. I am awere of how these computations work for the $A_n$ case (i.e., for flag varieties), ...
- 39.3k
2
votes
0
answers
158
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Standard definitions of some notions in algebraic geometry (canonical divisor, Q-Gorenstein, (log-)canonical/terminal, Fano, Calabi-Yau, General type)
I have a question about several related notions in algebraic geometry. I am mainly interested in the question "what is the standard notion?" (if there is such). But I also will be happy to ...
- 2,649
3
votes
1
answer
257
views
Reflections on affine quadric hypersurfaces
Let $f\colon\mathbb{Z}^n\otimes\mathbb{Z}^n\to\mathbb{Z}$ be a non-degenerate symmetric bilinear form and consider the affine quadric hypersurface $$
X:=\{f(x,x)+2=0\}\subseteq\mathbb{Z}^n.
$$
For ...
- 79.9k