All Questions
22,546 questions
1
vote
0
answers
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 ...
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 ...
- 2,928
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 ...
- 584
1
vote
0
answers
78
views
Invariant theory (first fundamental theorem) for a direct sum of two fundamental representations
Let $G$ be a simple reductive group over $\mathbb C$, e.g. $G=\mathrm{SO}(V)$ is a special orthogonal group.
Let $W_1$ and $W_2$ be two irreducible representations of $G$. Assume both $W_i$ are ...
- 6,622
3
votes
0
answers
164
views
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
0
answers
110
views
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
1
vote
0
answers
161
views
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
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 ...
- 43
1
vote
0
answers
117
views
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
9
votes
3
answers
699
views
I want to find a smooth section of the map from the Stiefel manifold to the Grassmanian manifold
The following question is related to research I am doing on reinforcement learning on manifolds.
I have a set of basis vectors $\boldsymbol{B} = \{\boldsymbol{b}_1,\dots,\boldsymbol{b}_k\}$ that span ...
- 155
1
vote
0
answers
113
views
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
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), ...
- 41
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
9
votes
2
answers
865
views
Multiplication in Peter-Weyl theorem
$\DeclareMathOperator\SL{SL}$It is known that the coordinate algebra $\mathcal O(\SL_n(\mathbb C))$ decomposes as direct sum of $V \otimes V^*$ for $V$ finite-dimensional irreducible representations ...
- 2,964
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 ...
- 119
2
votes
0
answers
92
views
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
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)
\...
- 807
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
0
votes
0
answers
127
views
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
0
votes
0
answers
99
views
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 ...
- 6,048
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 ...
- 6,048
1
vote
0
answers
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
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
5
votes
1
answer
290
views
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)$,
...
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?
- 393
0
votes
0
answers
123
views
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
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
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
2
votes
1
answer
198
views
Maximal sub-$\mathbb{C}$-algebras of $\mathbb{C}[x,y]$
After asking this question and finding this relevant paper, I would like to ask the following question:
For every $a,b \in \mathbb{C}$, denote:
$A_{a,b}=\mathbb{C}[(x-a)(x-b),x(x-a)(x-b),y]$
and
$B_{a,...
- 2,837
5
votes
2
answers
358
views
Canonical conics pulling back to polynomials on rational normal curve
(In following all schemes are formed over $\Bbb C$)
Let $C:=\nu_d(\Bbb P^1)$ the rational normal curve obtained via $d$-folded Veronese map $\nu_d: \Bbb P^1 \to \Bbb P^d$. The quadrics on $\Bbb P^d$ ...
- 6,048
3
votes
1
answer
320
views
Is the Hilbert Mumford Criterion true over the reals?
The Hilbert Mumford Criterion as in Wallach Theorem 3.24 says:
Let $G$ be a linearly reductive subgroup of $GL(n, \mathbb{C})$. Let $(\sigma, V)$ be a regular representation of $G$.
For a vector $v \...
- 161
2
votes
1
answer
224
views
Example of stable bundle whose pullback is polystable
Kempf (1992): "Pulling back bundles" has the following theorem:
Let $f: Y \rightarrow X$ be a finite morphism. If $\mathscr{W}$ is a bundle on $X$ that is stable with respect to an ample ...
- 577
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
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
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
0
votes
0
answers
48
views
Integral graded algebra of finite type is approximable
The following is the definition of approximable algebra.
An integral graded $K$-algebra $\oplus_{n\geqslant 0}B_n$ is said to be approximable if
1.$$rk_K(B_n)<+\infty,\forall n\in \mathbb{N}, $$and ...
- 1
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
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 ...
- 113
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
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} = \{...
- 125
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
0
answers
158
views
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
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
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
0
votes
2
answers
282
views
Can a variety be the graph of a function in more than one way?
Let $V\subset \Bbb R^n$ be an irreducible affine variety of degree $\ge 2$ and $U_V\subseteq V$ a (Euclidean) open subset. Suppose that $U_V$ is the graph of a rational function, that is, there is an ...
- 13.6k
3
votes
1
answer
253
views
About decomposition theorem BBD with respect to some stratification
I want to follow up a question from here (how to deduce version 1.a. from version 1).
I know a version of decomposition theorem BBD:
Version 1. Let $f:X\to Y$ be a (surjective) proper map of complex ...
- 133
4
votes
2
answers
585
views
Krull dimension in non-algebraically closed fields
Let $K$ be a field (not algebraically closed) and $F$ be its algebraic closure.
Let $X \subseteq K^n$ be Zariski closed, and $Y$ be the Zariski closure of $X$ inside $F^n$.
Is it true that $\dim(X) = \...
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
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
5
votes
1
answer
294
views
Compatibility of natural transformations in a six-functor formalism
Suppose we are given a six-functor formalism and a cartesian diagram
$$\require{AMScd} \begin{CD} X @>\tilde{g}>> Z \\ @V \tilde{f} V V @V Vf V \\ Y @>g>> W\end{CD} \,.$$
There are ...
- 913