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Results tagged with ag.algebraic-geometry
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user 1703
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
56
votes
3
answers
8k
views
What are the benefits of viewing a sheaf from the "espace étalé" perspective?
I learned the definition of a sheaf from Hartshorne—that is, as a (co-)functor from the category of open sets of a topological space (with morphisms given by inclusions) to, say, the category of sets. …
- 6,242
51
votes
3
answers
6k
views
What is the purpose of the flat/fppf/fpqc topologies?
There have been other similar questions before (e.g. What is your picture of the flat topology?), but none of them seem to have been answered fully.
As someone who originally started in topology/comp …
- 6,242
15
votes
Accepted
Is every algebraic $K3$ surface a quartic surface?
No. Consider a K3 surface with a polarization of degree 2 and with Picard rank 1. Since the tautological line bundle on $\mathbb{P}^3$ pulls back to a degree 4 line bundle, it follows that such a K3 s …
- 6,242
14
votes
0
answers
502
views
Am I missing something about this notion of Mirror Symmetry for abelian varieties?
This is a continuation of my recent question: Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s.
In the comments of the question, I was directed to the paper http://arxiv.org/abs/he …
- 6,242
13
votes
Accepted
Complex torus, C^n/Λ versus (C*)^n
The difference between an Abelian variety and $\mathbb{C}^n/\Lambda$ is that an abelian variety is polarized; that is, it comes with an ample line bundle, which yields an embedding into $\mathbb{P}^m$ …
- 6,242
12
votes
Elliptic curves on abelian surface
No. In general, there are no elliptic curves on an Abelian surface.
Thinking in terms of lattices, if there is an elliptic curve on $A$, then there is a rank 2 sublattice of $\Lambda$ (Where $A = \m …
- 6,242
12
votes
2
answers
1k
views
What classes am I missing in the Picard lattice of a Kummer K3 surface?
Constructing the Kummer K3 of an Abelian surface $A$, we have an obvious 22-dimensional collection of classes in $H^2(K3, \mathbb{Z})$ given by the 16 (-2)-curves (which by construction do not interse …
- 6,242
10
votes
1
answer
286
views
What can we say about tropical maps $\mathbb{P}^1 \to A$ for an Abelian variety $A$?
It's well known that maps $\mathbb{P}^1_\mathbb{C} \to A$ are constant for any Abelian variety $A$ (in fact, for any complex torus).
Is there any similar statement in the tropical case? Naively, the …
- 6,242
10
votes
1
answer
2k
views
What is the geometry behind psi classes in Gromov-Witten theory?
Intuitively, Gromov-Witten theory makes perfect sense. Via Poincare duality, we look at the cohomology classes $\gamma_1, \ldots, \gamma_n$ corresponding to geometric cycles $Z_i$ on a target space $X …
- 6,242
9
votes
3
answers
951
views
Is there an intrinsic way to define the group law on Abelian varieties?
On an elliptic curve given by a degree three equation y^2 = x(x - 1)(x - λ), we can define the group law in the following way (cf. Hartshorne):
We note that the map to its Jacobian given by $\mathca …
- 6,242
9
votes
1
answer
259
views
How is the propagator computed on an elliptic curve?
I've been struggling for a while now understanding why the propagator for the action
$$
S(\varphi) = \int_E \partial \varphi \bar\partial\varphi + \frac{\lambda}{6}(\partial\varphi)^3
$$
on an ellipti …
- 6,242
8
votes
1
answer
727
views
To what extent does Poincare duality hold on moduli stacks?
Poincare duality gives us, for a smooth orientable $n$-manifold, an isomorphism $H^k(M) \to H_{n-k}(M)$ given by $\gamma \mapsto \gamma \frown [M]$ where $[M]$ is the fundamental class of the manifold …
- 6,242
8
votes
0
answers
384
views
Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s
It is well known (cf. Dolgachev) that there is a beautiful notion of mirror symmetry for lattice-polarized K3 surfaces. That is, if we are given a rank $r$ lattice $M$ of signature $(1, r - 1)$ and a …
- 6,242
8
votes
What is the meaning of non-Hausdorff spaces in algebraic geometry
One of the things to think about in Algebraic Geometry is that the natural topology (Zariski topology) is the wrong topology, at least in part for the reasons you describe. There are other "topologies …
- 6,242
7
votes
1
answer
916
views
Trivial obstructions and virtual fundamental classes
Suppose $X$ is a DM stack, and let $E^\bullet$ be a perfect obstruction theory of $X$ such that the $E^{-1}$ term admits a trivial quotient/sub-bundle. Is it true that the virtual fundamental class $[ …
- 6,242
7
votes
2
answers
2k
views
What is a good reference (preferably thorough) for the Derived Category of a scheme/orbifold...
I've sort of circled around the idea of derived categories a few times, read a few introductory papers ("Derived Categories for the working mathematician", e.g.), and feel now that this is something t …
- 6,242
7
votes
Examples in mirror symmetry that can be understood.
I'm not sure if this is what you're looking for, but the paper "Mirror symmetry and Elliptic curves" by R. Dijkgraaf might be provide a good example.
The example in that paper concerns the mirror of …
- 6,242
6
votes
What is a branched Riemann surface with cuts?
Since no one has answered the genus computation, here goes:
The Riemann-Hurwitz formula states that for a map of Riemann surfaces $f : C_1 \to C_2$, that we have
$$
\chi(C_1) = n\chi(C_2) - \deg R
$ …
- 6,242
5
votes
Accepted
$\psi$ class in $\overline{M}_{0,n}$
Unless you mean something else when you write $\psi$ class, it is expressible in terms of boundary divisors.
That is, if $\psi_i$ is the $i$-th cotangent bundle, then you can write it in terms of bou …
- 6,242
5
votes
Accepted
Meaning of $g_d^r$ in algebraic geometry
As I understand, a $g_d^r$ is a linear system of dimeansion $r$ and degree $d$. Basically, these give you maps to $\mathbb{P}^r$ of degree $d$. The simplest example is of course hyperelliptic curves; …
- 6,242
4
votes
What is ample generator of a Picard group?
In terms of why these might be important, having an ample line bundle is equivalent to your object of study being projective. This is since, as in Henri's answer, ample implies that there in an embedd …
- 6,242
4
votes
Accepted
What is the geometric point of view of an algebraic line bundle compared to a analytic line ...
Perhaps this might help as some intuition. Instead of looking for "the line" in a locally free sheaf, let's look in the other direction. Let's start with a line bundle, and move back towards sheaves.
…
- 6,242
4
votes
Accepted
Trivial obstructions and virtual fundamental classes
It turns out that this is true. In the paper "Localizing Virtual Cycles by Cosections" by Kiem and Li, they address the case where one has a surjection $Ob \to \mathcal{O}$. In the case of an injectio …
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3
votes
Involution on Hyperelliptic curves and their Jacobians
Another way to think of this is the following (at least over $\mathbb{C}$).
Consider the diagram
$$
\begin{array}{ccccc}
X & \xrightarrow{AJ \times AJ \circ \sigma} & J \times J & & \\\\
\downarrow & …
- 6,242
3
votes
Accepted
Enumerativity of Gromov-Witten invariants of orbifolds
One simple example (although it is a genus 0 example) is the following.
Consider a global quotient $\mathscr{X} = [X/(\mathbb{Z}/2)]$. Then if we look at the genus 0 GW theory of $\mathscr{X}$ where …
- 6,242
3
votes
Accepted
Morphism between polarized abelian varieties
That should be true, yes.
A polarization of $A$ is given by a bilinear form on $H_1(A, Z)$; this is equivalent to a map $H_1(A,Z) \to H_1(A,Z)^\vee$, which is an isomorphism if the polarization is pr …
- 6,242
3
votes
Connection between 'Separated scheme of finite type over spec(k)' and 'Curve in $\mathbb R^n$
Well, first of all, a separated scheme of finite type over $Spec(k)$ is not necessarily a curve. A one dimensional separated scheme of finite type etc. etc. may be a curve, but this is also not quite …
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3
votes
What does it mean for a Deligne-Mumford stack to have trivial generic stabilizers?
The comment by user76758 hits the nail pretty much perfectly on the head. That said, it might be good to see an example of something that does not satisfy this condition:
Let $X$ be any scheme, and l …
- 6,242
2
votes
Accepted
"Restriction" of a fibre product to a subvariety
I believe this should be true: If you factor the map $\delta\mid_Y$ via the inclusion $Z \hookrightarrow D$ as
$$
Y \to Z \hookrightarrow D
$$
and pull back the map $\gamma$ along each of these maps, …
- 6,242
2
votes
on a Deformation long exact sequence of moduli space of stable maps
I don't believe that this is correct. The easiest way to see this is to look at your second question: The automorphisms/deformations/obstructions of a curve come from $H^i(C, T_C)$, i.e. they are the …
- 6,242
2
votes
2
answers
872
views
What is the difference between the moduli space of curves and the moduli space of orbi-curves?
Edit: In my original framing of this question it was not so clear what I was looking for, so this is basically a re-write.
I feel that I should already know the answer to this, but it never sits quit …
- 6,242
1
vote
Ramification of the map from the stack of elliptic curves to the $j$-line
I would guess that the following argument should work.
By definition, the map $\mathbb{H} \to \mathcal{M}_{1,1} = [SL_2\mathbb{Z} \setminus \mathbb{H}]$ is unramified. However, the map $\mathbb{H} \t …
- 6,242
1
vote
The Picard number of the Kummer surface of an abelian surface
Another way of phrasing this is to look at the transcendental lattices, which are the orthogonal complements of the Picard lattices.
One obtains in particular a morphism $T(A) \to T(Km(A))$ (by looki …
- 6,242
1
vote
Vanishing of the top Chern class of a vector bundle
The top Chern class is also the Euler class of the bundle, which is Poincaré dual to the homology class of the vanishing locus of a generic section. So these are equivalent.
- 6,242
1
vote
When do divisors pull back?
You shouldn't need to worry about whether or not the support intersects the closure $\overline{\varphi(X)}$. If it isn't, then your divisor simply pulls back to the zero divisor on X.
- 6,242
0
votes
Describing the kernel of the exponential map as a homology group
This morally sounds like it should be related to the exponential sequence in sheaf cohomology, but I don't immediately see how... Although, absent the torus (which I take as meaning $(\mathbb{C}^\time …
- 6,242
0
votes
0
answers
119
views
Obstruction theories on non-smooth spaces with smooth fibres
Given a perfect obstruction theory $E^\bullet$ over a space $X$, we know that if $X$ is smooth, that the virtual fundamental class $[X, E^\bullet]$ is given by
$$[X, E^\bullet] = c_{top}\big((E^{-1}) …
- 6,242