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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.
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
$$
\begin{array}{ccccc}
X & \xrightarrow{AJ \times AJ \circ \sigma} & J \times J & & \\\\
\downarrow & …
- 6,242
2
votes
Accepted
"Restriction" of a fibre product to a subvariety
I believe this should be true: If you factor the map $\delta\mid_Y$ via the inclusion $Z \hookrightarrow D$ as
$$
Y \to Z \hookrightarrow D
$$
and pull back the map $\gamma$ along each of these maps, …
- 6,242
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
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
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
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
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
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 …
- 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
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
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
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 …
- 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