All Questions
Tagged with examples ag.algebraic-geometry
79
questions
1
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0
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203
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Results that hold for the complex numbers but not for algebraically closed fields of characteristic zero
When a result is stated for the field of complex numbers it can usually be extended to a result for an algebraically closed field of characteristic zero. I would like to see a list of results that ...
- 147
2
votes
1
answer
117
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Example of maximal multicurve complex
in this paper we have :
" On the Teichmüller tower of mapping class groups By Allen Hatcher at Ithaca, Pierre Lochak at Paris and Leila Schneps."
Definition. The maximal multicurve complex $...
- 119
3
votes
0
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146
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Bounded derived categories of which smooth projectives possess bounded t-structures whose hearts are categories of modules?
I am interested in $P$ that is smooth and proper over a field and such that the derived category of coherent sheaves $D^b(P)$ possesses a $t$-structure whose heart is the category of finitely ...
- 16.3k
1
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0
answers
57
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Real (non-complex) Du Val singularities for quartics of high global Milnor number
I posted this in MathStackexchange and was advised to come here. As I am only looking for examples, I didn't feel MathOverflow was necessary.
I am looking for examples of specific quartic projective ...
- 111
2
votes
1
answer
201
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Making coherent sheaves with nonvanishing higher Chern classes
Let $\mathcal{F}$ be a coherent sheaf on a variety $X$, and assume $\mathcal{F}$ has generic rank $n$. I expect (see e.g. here) that this actually puts no conditions on its Chern classes $c_1(\mathcal{...
- 5,278
2
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0
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476
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description of very ample bundle of Hirzebruch surface
I learned some basic properties of Hirzebruch surface mainly from Vakil's notes "the rising sea", section 20.2.9. the Hirzebruch surface is defined as $\mathbb{F}_n:=\operatorname{Proj} (\...
- 181
4
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1
answer
365
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Push-out in the category of coherent sheaves over the complex projective plane
I'm trying to deal with an example of a rank two vector bundle over the complex projective plane which is non slope-stable (because the associated sheaf of sections has a coherent subsheaf of equal ...
- 385
1
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0
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126
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Examples for certain class of projective varieties
I am looking for specific varieties that satisfy certain property. I call them symmetric varieties, I want to know what varieties are in it. It contains the projective spaces $\mathbb{P}^n$ for all $n$...
- 5,791
0
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0
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218
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Symmetric products of varieties and projective bundles
Given a smooth projective geometrically connected curve $C$, a symmetric product of $C$ has the structure of a projective bundle over the Jacobian of $C$ (e.g. see Symmetric powers of a curve = ...
- 501
13
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1
answer
1k
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Explicit examples of Azumaya algebras
I'm trying to understand the Brauer group of a scheme better. I know how to compute $\text{Br}(X)$ as an abstract group in some cases, but don't have a good idea of what the individual Azumaya ...
- 5,278
7
votes
1
answer
457
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Example of a smooth family of projective surfaces with non-vanishing integrals of Todd classes
Motivation:
Let $\pi\colon S \rightarrow B$ be smooth projective morphism of relative dimension 2 over a smooth projective scheme $B$. If the stucture sheaves of the fibres do not have higher ...
- 76
19
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3
answers
2k
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For each $n$: show there is a genus $1$ curve over some field $k$ with no points of degree less than $n$, (simple argument / best reference)?
What is the simplest example (or perhaps best reference) for the fact that there are genus $1$ curves (over a field of your choice --- or if you wish, over $\mathbb{Q}$, to make it more exciting) with ...
- 3,837
5
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0
answers
217
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In search for examples concerning pushforward of nef divisors and lc-trivial fibrations
My question is motivated by ideas around the moduli b-divisor of an lc-trivial fibration (see for instance the following paper by Ambro https://arxiv.org/pdf/math/0308143.pdf).
In such a setup, one ...
- 625
40
votes
10
answers
4k
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Phenomena of gerbes
What is your favourite example of Gerbes?
I would like to know Where do we find Gerbes in "nature"?
The examples could vary from String theory to Galois theory. For example my favourite examples of ...
- 1,680
10
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1
answer
466
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Properties of the petit Zariski topos
What are some (intrinsically formulated) properties of the locally ringed topos $(\mathbf{Sh}(X),\mathcal{O}_X)$ for some scheme $X$, which do not hold for arbitrary locally ringed toposes?
Is there, ...
- 5,382
14
votes
1
answer
828
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Examples of étale covers of arithmetic surfaces
Define an arithmetic scheme $X$ to be a separated, integral scheme, flat and finite type over $\mathbb{Z}$. I am interested in obtaining examples of finite étale covers of arithmetic schemes. I am ...
- 1,338
74
votes
1
answer
5k
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$R$ is isomorphic to $R[X,Y]$, but not to $R[X]$
Is there a commutative ring $R$ with $R \cong R[X,Y]$ and $R \not\cong R[X]$?
This is a ring-theoretic analog of my previous question about abelian groups: In fact, in any algebraic category we may ...
19
votes
1
answer
747
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Vector field on a K3 surface with 24 zeroes
In https://mathoverflow.net/a/44885/4177, Tilman points out that one can use a $K3$ surface minus the zeroes of a generic vector field to build a nullcobordism for $24[SU(2)]$. Given that a) this is a ...
- 33.2k
4
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3
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850
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Examples of (non-normal) unibranched rings?
For a local integral domain $R$ the following are equivalent:
a) The integral closure of $R$ in its fraction field (i.e., the normalization of $R$) is again local.
b) The henselization of $R$ is ...
user78294
14
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0
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696
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Is Hironaka's example the only known deformation of Kähler manifolds with non-Kähler central fibre?
A well-known example in the deformation theory of compact complex manifolds is the one given by Hironaka in his 1962 paper An Example of a Non-Kählerian Complex-Analytic Deformation of Kählerian ...
- 18.5k
26
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4
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33k
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Recent, elementary results in algebraic geometry
Next semester I will be teaching an introductory algebraic geometry class for a smallish group of undergrads. In the last couple weeks, I hope that each student will give a one-hour presentation. ...
2
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145
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Is there a smooth rationally connected, proper variety $M$ over $\mathbb{C}$ such that c_1(L)([A]) = 1 which is not projective space?
Is there a smooth, rationally connected, projective variety $M$, which is not a projective space and is equipped with an ample line bundle $L$, such that the homology class of the curves $[A]$ which ...
- 3,608
1
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1
answer
343
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Examples of nontrivial local systems in Decomposition Theorem
There is a proper map $f: X \rightarrow Y$ of projective varieties. The Decomposition Theorem of Beilinson–Bernstein–Deligne-Gabber states that
$$Rf∗IC_X \cong \oplus_a IC_{\bar{Y_a}}(L_a)[shifts]$...
- 1,649
5
votes
2
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289
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smooth affine surfaces over algebraically closed fields with trivial l-torsion of the Brauer group
I am looking for examples of smooth affine surfaces over algebraically closed fields with trivial $\ell$-torsion of the Brauer group.
Related questions: Schemes with trivial brauer group and Brauer ...
user19475
4
votes
1
answer
315
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Unravelling some hypotheses on a variety
In Le group de Brauer II, Grothendieck states
Proposition 1.4.- Soit $X$ a préschéma noetherien. Supposon que les anneaux hensélisés stricts des anneaux locaux de $X$ soient factoriels, [...] Alors ...
- 33.2k
7
votes
3
answers
2k
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Fibrations with isomorphic fibers, but not Zariski locally trivial
(I posted this same question on MSE. Sorry if it is too elementary.)
I am looking for examples of fibrations $f:X\to Y$ where the fibers are all isomorphic, but $f$ is not Zariski locally trivial. In ...
- 1,514
1
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1
answer
516
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Examples of Quot schemes
I'm studying Quot schemes, that I denote with $Quot_{N,X,P}$, with $N \in \mathbb{Z}$, $X \subset \mathbb{P}^d$ and $P \in \mathbb{Q}[t]$. So, I'm looking for explicit examples of Quot schemes. Could ...
- 243
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1
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2k
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Learning a little Motivic Cohomology
Simply because I find it interesting, I have spent some time studying motivic cohomology from the lectures by Mazza, Voevodsky and Weibel. However, I'm finding it hard to tell if the theory is ...
- 3,495
1
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0
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564
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Homotopy theory of schemes
I have seen the notion of Homotopy come up in several contexts in schemes. For example, the book "Lectures on Motivic Cohomology" by Mazza, Weibel and Voevodsky uses this language to some extent. I.e. ...
- 33
4
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1
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437
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Clarification and intuition request for rationally equivalent algebraic cycles
I am having some difficulty lining up the definition and my intuition for rational equivalence of cycles. My intuition is based off of the idea that two cycles being rationally equivalent is analogous ...
- 43
4
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1
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460
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examples of Chow rings of surfaces
Can somone provide me (articles/literature) with examples of Chow rings of surfaces?
(e.g. here: http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf Chapter 9)
What I want is a list of (smooth ...
user19475
2
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1
answer
229
views
Non-uniruled variety with level one Hodge structure.
I wonder if there exists one example of non-uniruled algebraic variety with level one Hodge structure.
- 3,697
70
votes
28
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7k
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Examples where it's useful to know that a mathematical object belongs to some family of objects
For an expository piece I'm writing, it would be useful to have good examples of the following phenomenon:
(1) ${\cal X}$ is a parameterized family of somethings. (Varieties, schemes, manifolds, ...
32
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3
answers
3k
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Wanted: example of a non-algebraic singularity
Given a finitely generated $\def\CC{\mathbb C}\CC$-algebra $R$ and a $\CC$-point (maximal ideal) $p\in Spec(R)$, I define the singularity type of $p\in Spec(R)$ to be the isomorphism class of the ...
- 23.6k
7
votes
2
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1k
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The rank of a not necessarily finitely generated module.
This question is motivated by this one. The main point of the question (was) to try to weaken the notion of rank. After the answers and comments, it seems this is not a good way to do it, but perhaps ...
- 42.1k
6
votes
1
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2k
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An example of a rank one projective R-Module that is not invertible
Let $R$ be a commutative noetherian ring. I know that an $R$-module is invertible iff it is finitely generated and locally free of rank one. I presume then that there are examples of non-finitely ...
- 325
62
votes
6
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5k
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Simplest examples of nonisomorphic complex algebraic varieties with isomorphic analytifications
If they are not proper, two complex algebraic varieties can be nonisomorphic yet have isomorphic analytifications. I've heard informal examples (often involving moduli spaces), but am not sure of the ...
- 3,837
13
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2
answers
1k
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Families of curves for which the Belyi degree can be easily bounded
I know (edit: three) families of smooth projective connected curves over $\bar{\mathbf{Q}}$ for which the Belyi degree is not hard to bound from above.
The modular curves $X(n)$. They are ...
- 9,377
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6
answers
4k
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Examples of naturally occurring Quadratic forms or quadrics.
I am always fascinated when a quadratic form (or a quadric) arises naturally. I have
some elementary examples, but most of all, I want to learn more examples. I hope this question isn't considered too ...
10
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3
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388
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Algebraic Curves and Phase Diagrams of Physical Systems
Lots of low degree curves arise naturally as the phase spaces of physical systems (that is, the curve parameterized by $(q,p)$ where $q$ is a generalized position variable and $p$ is a generalized ...
- 15.7k
25
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2
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2k
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Examples where the analogy between number theory and geometry fails
The analogy between $O_K$ ($K$ a number field) and affine curves over a field has been very fruitful. It also knows many variations: the field over which the curve is defined may have positive or zero ...
4
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0
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488
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Example of a Grothendieck pretopology satisfying a weak saturation condition
Recall that a singleton Grothendieck pretopology (henceforth 'singleton pretopology') on a category $C$ is a collection of maps $J$ containing the isomorphisms, closed under composition and stable ...
- 33.2k
2
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0
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1k
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Example of smooth, proper but non-projective curve over an affine, connected base?
Would someone please give an example of a smooth, proper but non-projective curve $C/S$, where $S$ is affine and connected? I believe that whatever your example, $C/S$ must have genus $1$, admit no ...
- 2,220
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1
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215
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Geometric explanation of an orbit space: Integer action on the affine line
Let $k$ be a field of char $0$ and let $\mathbb{Z}$ act on $\mathbb{A}^1_k$ by the action induced by $G\to\mathrm{Aut}_k(k[X]), n\mapsto X+n$. It is rather easy to show that the orbit space $\mathbb{A}...
- 1,233
4
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1
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3k
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Ringed and locally ringed spaces
A pair $(X,O_X)$ is a ringed space if $X$ is a topological space and $O_X$ is a sheaf of rings. If every stalk $O_{X,x}$ is a local ring, then we say that $(X,O_X)$ is a locally ringed space.
In the ...
- 2,175
8
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3
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864
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Is $\mathbb{A}²$ the universal smooth scheme which is a finite cover of $\mathbb{A}²/μ₂$?
One very handy (counter)example I often think about is the scheme $Spec(k[a,b,c]/(ab-c^2))$ (where $k$ is a field), which you may also know as $Spec(k[x^2,xy,y^2])$, as $\mathbb A^2/\mu_2$, or as the $...
- 23.6k
78
votes
23
answers
18k
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Algebraic geometry examples
What are some surprising or memorable examples in algebraic geometry, suitable for a course I'll be teaching on chapters 1-2 of Hartshorne (varieties, introductory schemes)?
I'd prefer examples that ...
19
votes
4
answers
2k
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Example of a smooth morphism where you can't lift a map from a nilpotent thickening?
Definition. A locally finitely presented morphism of schemes $f\colon X\to Y$ is smooth (resp. unramified, resp. étale) if for any affine scheme $T$, any closed subscheme $T_0$ defined by a square ...
- 23.6k
5
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0
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239
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Is the field of invariants $k(V)^G$ purely transcendental over $k$?
Reference: http://www.math.u-psud.fr/~colliot/mumbai04.pdf
Proposition 4.3. on page 18 in the above reference reads as follows:
Assume $k = \overline{k}$. If $V$ is a finite dimensional vector space ...
- 2,742
2
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1
answer
1k
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Example of restriction of a finite morphism which is not finite
Every closed immersion is a finite morphism. Therefore, restriction of a finite morphism to a closed subset is always a finite morphism itself. Can you give an example of quasi-projective varieties $X\...
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