3
votes
3answers
477 views

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 ...
2
votes
1answer
277 views

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 ...
14
votes
1answer
750 views

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 ...
1
vote
0answers
274 views

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. ...
2
votes
1answer
212 views

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 ...
3
votes
1answer
233 views

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 ...
2
votes
1answer
168 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.
40
votes
27answers
4k views

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, ...
26
votes
3answers
2k views

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 ...
5
votes
2answers
626 views

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 ...
4
votes
1answer
1k views

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 ...
45
votes
6answers
3k views

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 ...
10
votes
2answers
847 views

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 ...
4
votes
2answers
2k views

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
votes
3answers
345 views

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 ...
21
votes
2answers
2k views

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
votes
0answers
269 views

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 ...
1
vote
0answers
531 views

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 ...
0
votes
1answer
167 views

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 ...
1
vote
1answer
731 views

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 ...
6
votes
3answers
640 views

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 ...
15
votes
4answers
1k views

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 ...
2
votes
1answer
574 views

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 ...
1
vote
1answer
422 views

Example of inclusion which is not a finite morphism [closed]

Every closed immersion is a finite morphism. Can you give an example of quasi-projective varieties $X\subset Y$ such that inclusion $X\hookrightarrow Y$ is not finite? Same with Y projective? Thanks! ...
10
votes
2answers
2k views

non principally polarized complex abelian varieties

I've read in (abstracts of) papers that there are abelian varieties over fields of positive characteristic that admit no prinicipal polarization. Apparently its not the easiest thing to find an ...
2
votes
0answers
239 views

Forgetting extra structure inducing Symmetries

This is a major edit of the original post after receiving helpful comments. It is often the case when one adds additional structure to make a problem more tractable. When one attempts to forget this ...
4
votes
4answers
491 views

Kähler manifold which is not algebraic

Can someone provide examples of Kähler manifolds which are not algebraic? This question came to my mind seeing the post of Andrea Ferretti.
10
votes
8answers
996 views

What are examples illustrating the usefulness of Krull (i.e., rank > 1) valuations?

In modern valuation theory, one studies not just absolute values on a field, but also Krull valuations. The motivation is easy enough: If $k$ is a field, a valuation ring of $k$ is a subring $R$ ...
7
votes
4answers
743 views

Examples of Poisson Schemes

A Poisson Manifold is a real manifold $M$ along with a Lie bracket $[\cdot,\cdot]$ on $C^\infty(M)$ which is a derivation in each variable. Poisson manifolds are interesting for a few reasons, among ...
5
votes
1answer
418 views

obstruction to smooth lifting of smooth schemes

According to general theory, for a square zero thickening defined by an ideal I: SpecA -> SpecA', there is an obstruction of lifting a smooth scheme X over A to a smooth scheme over A' living in ...
2
votes
4answers
708 views

Examples of divisors on an analytical manifold

I am trying to understand divisors reading through Griffith and Harris but it is difficult to come up with any particular interesting example. I have browsed through Hartshone's book but everything is ...
9
votes
4answers
1k views

Negative Gromov-Witten invariants

I understand the heuristic reason why Gromov-Witten invariants can be rational; roughly it's because we're doing curve counts in some stacky sense, so each curve $C$ contributes $1/|\text{Aut}(C)|$ to ...
15
votes
3answers
1k views

Algebraic varieties which are topological manifolds

Inspired by this thread, which concludes that a non-singular variety over the complex numbers is naturally a smooth manifold, does anyone know conditions that imply that the topological space ...
5
votes
3answers
596 views

Gerbes for a cyclic group. (or maybe G_m too)

Let μn be the group scheme of n-th roots of unity. If X is a scheme and L is a line bundle on X, then I can construct a μn-gerbe Y over X by letting the S-points of Y be a S-point of X, a line ...
12
votes
0answers
528 views

Pencils with many completely decomposable fibers

Let $F= \frac{G}{H} : \mathbb P^n \to \mathbb P^1$ be a non-constant rational function ($G$ and $H$ homogenous polynomials of the same degree in $\mathbb C^{n+1})$. The fiber over $(\lambda:\mu) \in ...
8
votes
1answer
317 views

Can an algebraic space fail to have a unviersal map to a scheme?

Let $\mathcal{X}$ be an algebraic space. Can it happen that there does not exist a map $\mathcal{X} \to X$ with $X$ a scheme that is initial for maps from $\mathcal{X}$ to schemes? Are there ...
6
votes
3answers
650 views

Examples of birational equivalence of a variety and a hypersurface

There's an algebraic geometry theorem (I.4.9 in Hartshorne) that says: any variety of dimension r (over an algebraically closed field) is birationally equivalent to a hypersurface in projective space ...
7
votes
3answers
909 views

Nonprojective Surface

Let k be an algebraically closed field. It's well known that every complete curve, period, is projective. Also, that every smooth surface is, and that there are smooth 3-folds which are not, and ...
6
votes
0answers
544 views

“a theory of generalized Donaldson-Thomas invariants” by Joyce & Song

Is anyone else working through this paper : http://arxiv.org/abs/0810.5645 ? I am trying to verifying example 6.2 (m=2 for simplicity) using only the definitions, namely: J^{2\alpha}(\tau) = -1/4 ...
13
votes
5answers
1k views

What are the most important instances of the “yoga of generic points”?

In algebraic geometry, an irreducible scheme has a point called "the generic point." The justification for this terminology is that under reasonable finiteness hypotheses, a property that is true at ...
10
votes
5answers
3k views

Examples and intuition for arithmetic schemes

How should a beginner learn about arithmetic schemes (interpret this as you wish, or as a regular scheme, proper and flat over Spec(Z))? What are the most important examples of such schemes? Good ...
2
votes
5answers
735 views

Is very ampleness of a divisor on a curve determined entirely by degree and genus?

Edit: Apparently the answer is "no", so what is an example of two curves of genus g, and a divisor of degree d on each, such that one is very ample and the other is not? Question as originally ...
6
votes
1answer
464 views

Example where you *need* non-DVRs in the valuative criteria

The valuative criterion for separatedness (resp. properness) says that a morphism of schemes (resp. a quasi-compact morphism of schemes) f:X→Y is separated (resp. proper) if and only if it ...
13
votes
7answers
2k views

Examples of rational families of abelian varieties.

I'd like to know examples of non-trivial families of abelian varieties over rational bases (e.g. open subschemes of the projective line P^1). One can generate many examples as Jacobians of rational ...
13
votes
3answers
804 views

Is there an example of a variety over the complex numbers with no embedding into a smooth variety?

Is there an example of a variety over the complex numbers with no embedding into a smooth variety?
6
votes
1answer
382 views

Can one check formal smoothness using only one-variable Artin rings?

Let f:X -> Y be a morphism of schemes over a field k. Can one check that f is formally smooth using only Artin rings of the form k'[t]/t^n, where k' is also a field? Considering cuspidal curves one ...
11
votes
1answer
614 views

Is there an example of a scheme X whose reduction X_red is affine but X is not affine?

For Noetherian schemes this follows from Serre's criterion for affineness by a filtration argument.
11
votes
3answers
899 views

Can a coequalizer of schemes fail to be surjective?

Suppose $g,h:Z\to X$ are two morphisms of schemes. Then we say that $f:X\to Y$ is the coequalizer of $g$ and $h$ if the following condition holds: any morphism $t:X\to T$ such that $t\circ g=t\circ h$ ...
5
votes
3answers
377 views

Weil divisors on non Noetherian schemes

Let X be an integral scheme that is separated (say over an affine scheme). Define a Weil divisor as a finite integral combination of height 1 points of X, where the height of a point of X is the ...
5
votes
3answers
475 views

If \Omega_X/Y is locally free of rank dim(X)-dim(Y), is X->Y smooth?

Suppose I have a morphism f:X→Y such that the relative sheaf of differentials ΩX/Y is locally free. Does it follow that f is smooth? The answer is no, but for a silly reason. You could ...