All Questions
22,546 questions
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
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 ...
- 6,038
0
votes
0
answers
44
views
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
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{...
- 199
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 $Y$ ...
- 6,038
1
vote
1
answer
154
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
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 ...
- 341
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 ...
- 6,038
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
41
votes
1
answer
2k
views
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 ...
- 213
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
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 ...
- 127
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
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
2
answers
218
views
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. ...
- 156k
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
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
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 $D^b(\mathbb{P}^n) \hookrightarrow D^b(\mathbb{P}^N)$?
- 123
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
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
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
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 ...
- 6,038
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
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
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
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
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}}, \...
- 333
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
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 ...
- 1
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
4
votes
0
answers
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
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
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
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
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 ...
- 9,019
1
vote
0
answers
42
views
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
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
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
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
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
1
vote
1
answer
291
views
General algebraic definition of mirror symmetry
I'm trying to understand the following statement of Hori-Vafa from the algebraic perspective:
The mirror of the Hirzebruch surface $\mathbb{F}_{n}$ is the Landau-Ginzburg model $x+y+\frac{a}{x}+\frac{...
- 305
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
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
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 \...
- 71
6
votes
1
answer
407
views
Good reduction for the universal elliptic curve
Let $X$ be a modular curve, i.e. a quotient of the upper half plane $\mathbb{H}$ by a congruence subgroup $\Gamma$. When $\Gamma=\Gamma_0(N)$, it is known that $X$ has a smooth model denoted $\mathcal{...
- 2,907
3
votes
0
answers
156
views
A possible application of deformation theory?
Let $f : \mathbb{R}^{n} \to \mathbb{R}$ be a real-valued polynomial function. Consider the family of real algebraic sets:
$$
V_c = f^{-1}(c), \quad c \in (-1,1).
$$
I am interested in determining how ...
- 357
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
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
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