All Questions
22,546 questions
0
votes
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 ...
- 6,048
14
votes
1
answer
3k
views
An elementary proof that the degree of a map of spheres determines its homotopy type
I'm helping to teach an undergraduate algebraic topology course (out of Hatcher's textbook). We have recently defined the degree of a map of spheres using homology, and the professor and I thought it ...
- 486
0
votes
0
answers
78
views
Is the torus of any affine, simplicial toric variety always split?
Let $k$ be an arbitrary field and $X$ be an affine, simplicial toric variety over $k$ of dimension $n$. Then $X$ has the form $\operatorname{Spec}(k[\sigma^{\vee}\cap\mathbb{Z}^n])$ for some $n$-...
- 639
1
vote
0
answers
71
views
Computing the symmetric product of a sheaf supported on a divisor
Let $Y\hookrightarrow X$ be a inclusion of a smooth divisor $Y$ (i.e. codimension 1 closed subscheme) in to smooth variety $X$. Let $\mathcal{L}$ be a vector bundle over $X$, we regard it as a locally ...
- 653
42
votes
13
answers
20k
views
How to draw knots with LaTeX?
I am writing an exam for my students, and the topic is intro knots theory. I have no idea how to put knots into the file, but I know many MO users who can draw amazing diagrams in their papers.
Can ...
- 12.9k
1
vote
0
answers
139
views
Estimation of the degree of a projective surface
Let $S \subset \mathbb{P}^{n}_{\mathbb{C}}$ a projective integral smooth surface lying in no hyperplane (I adopt the point of view of scheme). I will denote by $d$ its degree. The following questions ...
- 73
4
votes
1
answer
326
views
Is the pushforward of a closed immersion ever fully-faithful at the level of Derived Categories?
Let $i: Z \rightarrow X$ be a closed immersion of schemes. Then, for any $\mathcal{O}_{Z}$-module $\mathcal{G}$, the counit of adjunction $i^{*}i_{*}\mathcal{G} \rightarrow \mathcal{G}$ is an ...
- 15.9k
5
votes
1
answer
568
views
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 ...
- 56
0
votes
0
answers
125
views
Néron-Tate height on abelian varieties and PDEs
Let $A$ be an abelian variety over some number field $K$. We know the $\mathbb{C}$-points of $A$ form a complex analytic manifold, so $A(\mathbb{C})$ is a smooth manifold in fact. It then makes sense ...
1
vote
1
answer
133
views
Can we find a Jouanolou device for $\mathbb{P}^d$ having dimension $<2d$?
Let us work over an algebraically closed field $k$.
A Jouanolou device for a $k$-variety $X$ is an affine space fiberation $f:Y\to X$ such that $Y$ is an affine scheme. (The condition on $f$ means ...
- 921
1
vote
0
answers
116
views
Universal picard variety of degree d
Let $M_g$ denote the moduli space of smooth genus $g$ curves over $\mathbb{C}$, where $g \geq 2$. Let $Pic^{d,g}$ denote the universal picard variety over $M_g$, parameterizing pairs $(C,L)$ where $C$...
- 129
1
vote
0
answers
105
views
Isomorphism between spectrum of $\mathcal{O}_{\mathbb{P}^1_{[y_0:y_1]}}[y_0^2t,y_0y_1t,y_1^2t]$ and the line bundle $\mathcal{O}_{\mathbb{P}^1}(2)$
Let $\mathbb{P}^1$ be the projective line over a base field $k$, with homogeneous coordinates $[y_0 : y_1]$. Consider the sheaf of $\mathcal{O}_{\mathbb{P}^1}$-algebras $\mathcal{A} = \mathcal{O}_{\...
- 11
5
votes
1
answer
336
views
Counterexample to flat base change for $\mathcal{O}_X$-modules
Consider a cartesian diagram
$$\require{AMScd}
\begin{CD}
X' @>{f'}>> X\\
@V{p'} VV @VV{p} V\\
S' @>{f}>> S
\end{CD}$$
of schemes (or even locally ringed spaces). If $\mathcal{F}$ is ...
- 921
6
votes
1
answer
189
views
Fully faithful embeddings of derived category of projective space into derived category of a higher dimensional projective space
Let $N>n$. Are there are any known cases of a fully faithful embedding $D^b(\mathbb{P}^n) \hookrightarrow D^b(\mathbb{P}^N)$?
- 39.3k
3
votes
0
answers
169
views
equivalence of two categories
I am new to algebraic geometry and category theory. I am wondering about the following functor is equivalence of categories or not.
Let $X$ be irreducible scheme and $x$ be its unique generic point. ...
- 613
11
votes
1
answer
592
views
Degree inequality of a polynomial map distinguishing hyperplanes
Let $H_1, \ldots, H_n$ be $n$ linearly independent hyperplanes in $k^n$, for some arbitrary field $k$. Let $X = H_1 \cup H_2 \cup \cdots \cup H_n$. Is it true that if $F=(f_1, \ldots, f_n)$ is a ...
CommunityBot
- 1
5
votes
1
answer
212
views
Stability of ODEs with polynomial nonlinearity
Consider the following ODE system:
$$
x′=f(x)\iff
\begin{pmatrix}
x_1^\prime \\
\vdots\\
x_k^\prime\\
\vdots\\
x_n ^\prime
\end{pmatrix} =
\begin{pmatrix}
f_1(x) \\
\vdots\\
f_k(x)\\
\vdots\\
f_n(x)
\...
- 141
5
votes
0
answers
252
views
Does a simple formal group give rise to a simple Lie algebra?
A formal group $F$ is an intermediate object between Lie group $G$ and Lie algebra $\mathfrak{g}$.
A formal group is called simple if it has no nontrivial sub-formal group. On the other hand, a simple ...
- 195
1
vote
0
answers
129
views
Is $K_0(\mathrm{Vect}(X))\to K_0'(X)$ injective for a proper variety $X$?
Let $X$ be an integral scheme, proper over an algebraically closed field $k$. Let $\mathrm{Vect}(X)$ be the exact category of finite locally free $O_X$-modules. Let $K_0(\mathrm{Vect}(X))$ be its ...
- 615
2
votes
1
answer
192
views
Does Serre's condition $S_k$ depend only on codimension $\leq k$ points?
Recall a locally Noetherian scheme $X$ has Serre's condition $S_k$ if for every $x\in X$ we have $\mathrm{depth}(\mathcal{O}_{x,X})\geq \mathrm{min}(k,\mathrm{dim}(\mathcal{O}_{x,X}))$.
Let $X$ be a ...
- 4,111
7
votes
0
answers
151
views
Discriminants and lattices in Algebraic geometry vs Geometry of numbers
(Post-writing, this question ended up being way more rambly than I intended. Sorry for that. There's a lot of closely related ideas I'm trying to unravel and it's hard to extract an individual ...
- 473
5
votes
1
answer
367
views
Check that a Sheaf is Invertible Etale Locally
A question about following statement from Martin Olsson's book on Stacks. In the proof of Proposition 13.2.9. (p 269) is claimed that certain sheaf $K$ on a nodal curve $C$ is invertible it suffice to ...
- 4,008
2
votes
2
answers
587
views
Hilbert scheme of divisors in smooth projective varieties
Let $X$ be a smooth projective variety and $L$ be a line bundle with $H^0(X,L)\neq 0$. Let $D\in |L|$ and $p(t)$ be the Hilbert polynomial of $D$. Assume that any effective divisor $D'\subset X$ with ...
- 4,111
1
vote
0
answers
102
views
weak (?) valuative criterion for properness
In the article "On the Kodaira Dimension of the Moduli Space of Curves" by J. Harris and D. Mumford, to prove that
$\overline{H}_{k,b}$ is proper over Spec $\mathbb{C}$, the authors refer to ...
- 560
2
votes
1
answer
505
views
Family of curve singularities whose generic members are smooth
Let $f: (X,x)\rightarrow (\mathbb C,0)$ be a deformation of a curve singularity $(X_0,x)$, and let $f: X \rightarrow T$ be a sufficiently small representative. Assume that $(X,x)$ is reduced and pure ...
CommunityBot
- 1
6
votes
0
answers
554
views
What remains true after condensation?
As I slowly develop intuition for the condensed formalism, I feel that it’d help greatly if there were a principle or meta-theorem which said that certain kinds of statements that are true in ordinary ...
- 217
7
votes
1
answer
626
views
Action of the symmetric group $S_3$ on an elliptic curve $E$ defined over $\mathbb{Z}$
I came up with the following question on a facebook group: find the positive integer solutions of the equation $$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=4$$
Now clearly this is very difficult, ...
- 2,858
8
votes
1
answer
653
views
Status of a conjecture in Grothendieck's "Crystals and the de Rham Cohomology of Schemes"
Let $X/\mathbb{C}$ be a scheme over the complex numbers. In "Crystals and the de Rham cohomology of schemes," Grothendieck constructs the infinitesimal ringed site $(X_{\operatorname{inf}}, \...
- 2,856
1
vote
1
answer
235
views
Is the Cox ring of a Mori dream space $ Z $ which is of globally $ F $-regular type Cohen Macaulay?
A variety $ X $ over a field of characteristic zero is of globally $ F $-regular type if there is a ring $ A $ which is a finitely generated $ \mathbb{Z} $-algebra, a flat family
$ \mathcal{X} $ over $...
CommunityBot
- 1
4
votes
4
answers
1k
views
Degree of a variety is well-defined
Let $X$ be a projective variety embedded in a projective space, over a field of arbitrary characteristic.
What is a good reference for a nice proof of the classic fact that degree of $X$ is well-...
- 13.6k
3
votes
0
answers
404
views
Jacobians of curves with maximal Picard number
What can be said about a complex curve $C$, if its jacobian $J(C)$ has the maximal Picard number?
It is natural to expect that for a general curve of given genus its Jacobian has Picard rank 1 (isn't ...
CommunityBot
- 1
6
votes
0
answers
223
views
Properties of cohomology stacks
For several cohomology theories for schemes it is possible to construct a geometric model: for any suitable scheme $X$ it is a ring stack $\mathcal{H}(X)$, defined over the coefficient ring $R$ of the ...
- 217
8
votes
0
answers
405
views
Motives and ring stacks
In the lecture “Motives and ring stacks” Peter Scholze begins by saying that there are many ‘Weil-type’ cohomology theories for (separable and finite type) schemes over $\mathbb{Z}$ each of which can ...
- 217
0
votes
0
answers
95
views
Conditions for regularity in a covering
Let $V$ be a DVR of mixed characteristic, whose residue field is a finite field of characteristic $2$. Let $R$ be a flat, finitely generated algebra over $V$, which is regular. Let $a\in R^*$ be an ...
- 4,713
1
vote
0
answers
73
views
Is the subgroup of $R$-trivial classes of an algebraic group an algebraic subgroup?
Let $G$ be an algebraic group over a field $F$. I'm willing to assume it is linear if that changes anything to what I'm going to say, and even reductive if that helps (but I don't think it should make ...
- 272
2
votes
0
answers
109
views
Punctured neighbourhood of quotient singularity is not simply connected?
Let $X$ be a variety (irreducible, normal) over a field $k$ which is algebraically closed with characteristic $0$. Suppose that $X$ has only one singular point $p\in X$, so $Y:=X\setminus p$ is smooth....
- 131
4
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0
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97
views
Is there a concept of a map of Grothendieck sites having dense image?
Someone recently asked if one can talk about a map being etale dense just like one can talk about it being Zariski dense. My main question is: has anyone discussed such a notion?
On a simple ...
- 15.4k
1
vote
1
answer
238
views
Generalization of Gromov-Witten theory counting surfaces, 3-folds, etc
I don't work on the Gromov-Witten theory, but I find that I need to study the following problem, which seems to be similar to the Gromov-Witten theory:
Let $X$ be a variety and $\alpha_{1}, \cdots, \...
CommunityBot
- 1
2
votes
0
answers
129
views
Is the deformation of a $C^{\infty}$-manifold over Artin local algebra trivial?
$\DeclareMathOperator\Spec{Spec}$Let $X$ be a compact $C^{\infty}$-manifold without boundary. Let $(A,m)$ be a Artin local $\mathbb{C}$-algebra such that $A/m\cong \mathbb{C}$. Intuitively, a ...
- 63.9k
3
votes
1
answer
163
views
Is a pseudo-effective divisor on a rational surface numerically effective?
Let $D$ be a pseudo-effective $\mathbb{R}$-divisor on a rational surface. Can we find an example that the numerical class of $D$ contains no effective divisor?
- 549
28
votes
2
answers
2k
views
How did Riemann prove that the moduli space of compact Riemann surfaces of genus $g>1$ has dimension $3g-3$?
Consider the moduli space $M_g$ of compact Riemann surfaces (i.e., smooth complete algebraic curves over $\mathbb{C}$) of genus $g$ for some $g>1$. I'm interested in knowing how Riemann proved that ...
- 91.8k
1
vote
0
answers
42
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Example polynomial system where Macaulay bound is tight
I have been solving systems of polynomial equations by forming the Macaulay matrix of different degrees and computing its null space. If the degree is large enough, namely at or above the degree of ...
- 11
2
votes
0
answers
123
views
Isomorphism between motivic cohomology and algebraic cobordism
Let $MGL$ be the algebraic cobordism defined by Voevodsky, and $\Omega$ the algebraic cobordism constructed by Levine and Morel. For motivic cohomology $H^{p,q}$, we use Suslin-Voevodsky's definition. ...
- 121
3
votes
1
answer
302
views
Local acyclicity with respect to a sheaf
I am trying to understand the definition 2.12 in SGA 4.5, chapter 7. The multitute of localizations confuses me, as I still do not understand very basic algebraic geometric notions.
To get some ...
- 12.9k
5
votes
1
answer
633
views
Consistency of ZFC with inaccessible cardinals but no measurable cardinals
Let $S$ be a set and k a infinite field. The injection $S \to k\mathrm{Alg}(k^S, k)$ (sending a point to the evaluation in it) is a bijection if and only if $S$ is a non-measurable cardinal (see for ...
- 6,962
4
votes
1
answer
236
views
If a lattice can be embedded into $\mathbb Q^n,\langle-1\rangle^n$, then can it be embedded into $\mathbb Z^n,\langle -1 \rangle^n$?
Given a graph with negative integers on each vertex $\Gamma$ there is a corresponding intersection lattice denoted $Q_\Gamma$, a free $\mathbb Z$ module generated by the vertices, endowed with a ...
- 12.9k
4
votes
1
answer
284
views
Are the two definitions of fppf topology on the category of schemes the same?
Consider the definition of fppf (pre)topology on the category of schemes $\mathrm{Sch}$.
Maybe, most textbooks define an fppf covering of $U\in\mathrm{Sch}$ as a family of morphisms $\mathscr{U} = \{...
- 15.9k
1
vote
1
answer
249
views
Higher cohomology of line bundles and small modifications
I am a PhD student in algebraic geometry and I am blocked on a question that I asked to myself for my research. I am not yet very comfortable with the traditional tools. Also, I apologize in advance ...
- 39.3k
4
votes
0
answers
214
views
Algebraic logic in the style of algebraic geometry
I am writing a thesis on algebraic logic, I wonder if there is any recent research on an idea mentioned in Yuri Manin's book on algebraic geometry and in another Russian textbook on differential ...
- 558
0
votes
0
answers
104
views
Non-degenerate bilinear pairing of finite dimensional algebras
A finite dimensional algebra (over $\mathbb{C}$, say) is said to be Frobenius if it comes equipped with a nondegenerate bilinear pairing
\begin{align*}
\langle -, - \rangle : A \times A \rightarrow \...
- 63.9k