All Questions
22,770 questions
19
votes
3
answers
5k
views
What is the relationship between algebraic geometry and quantum mechanics?
The basic relationship in algebraic geometry is between a variety and its ring of functions. Arguably a similarly basic relationship in quantum mechanics is between a state space and its algebra of ...
- 118k
35
votes
5
answers
10k
views
The Relationship between Complex and Algebraic Geomety
I have recently begun to study algebraic geometry, coming from a differential geometry background. It seems that there is a deep link between complex manifolds and complex varieties. For example, one ...
- 3,399
9
votes
1
answer
399
views
Existence of hyperelliptic curve with specific number of points in a family
Hi,
the following question was posed to me, it apparently has applications for linear codes. Let n>1, and $K = \rm{GF}(2^n)$. Let $k$ be coprime to $2^n-1$. Does there always exist $a \neq 0$ in $K$ ...
- 40.2k
1
vote
3
answers
2k
views
Various Cartan's Lemmata
I am a bit amazed by "Cartan's Lemma".. I have so far seen it in :
Algebraic Geometry sources:
Look at Proposition 2.9 of Freitag and Kiehl's Étale Cohomology where he used étale morphism to describe ...
- 2,275
7
votes
1
answer
258
views
39
votes
3
answers
6k
views
What do higher Chow groups mean?
Let $z^i(X, m)$ be the free abelian group generated by all codimension $i$ subvarieties on $X \times \Delta^m$ which intersect all faces $X \times \Delta^j$ properly for all j < m. Then, for each i,...
- 12.3k
5
votes
2
answers
953
views
Singular K3 -- mathematical meaning?
There's a very interesting text by Cumrun Vafa called Geometric Physics.
Here I'm particularly interested in Chapter 4, where we take a Calabi-Yau manifold presented as a degenerating fibration:
...
- 15.1k
16
votes
4
answers
1k
views
K3 surfaces with good reduction away from finitely many places
Let S be a finite set of primes in Q. What, if anything, do we know about K3 surfaces over Q with good reduction away from S? (To be more precise, I suppose I mean schemes over Spec Z[1/S] whose ...
- 19.2k
12
votes
3
answers
2k
views
level sets of multivariate polynomials
Let $p:\mathbb R^n \rightarrow \mathbb R$ be a polynomial of degree at most $d$. Restrict $p$ to the unit cube $Q=[0,1]^n\subset\mathbb R^n$. We assume that $p$ has mean value zero on the unit cube $Q$...
- 621
16
votes
3
answers
2k
views
The Sylvester Gallai Theorem and Sections of Varieties with "Simple Topology".
The Sylvester-Gallai theorem asserts that for every collection of points in the plane, not all on a line, there is a line containing exactly two of the points.
One high dimensional extension ...
19
votes
9
answers
4k
views
Hironaka desingularisation theorem -- new proofs in literature?
I'm wondering what the landscape looks like for proofs of Hironaka's desingularisation theorem.
Are there many proofs in the literature?
Is there a commonly accepted simplest bare-knuckle proof out ...
- 44.3k
15
votes
1
answer
668
views
Can an algebraic space fail to have a universal 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 ...
- 10.5k
7
votes
1
answer
1k
views
Valuative criterion for properness
Let $f : X \rightarrow Y$ be a finite type morphism of Noetherian schemes. The valuative criterion for properness runs as follows. Suppose that for any DVR $R$ with fraction field $K$ that any $K$-...
- 1,209
27
votes
3
answers
5k
views
When is an algebraic space a scheme?
Sometimes general theory is "good" at showing that a functor is representable by an algebraic spaces (e.g., Hilbert functors, Picard functors, coarse moduli spaces, etc). What sort of general ...
- 1,990
23
votes
5
answers
10k
views
Maps to projective space determined by a line bundle
The following should be pretty standard for any algebraic geometer.
Let $X$ be a compact complex variety, and let $L$ be a line bundle on $X$. We say $L$ is 'generated by global sections' if for ...
- 13k
6
votes
1
answer
1k
views
Uniformization in algebraic/arithmetic geometry?
Jonah's question makes me wonder: What is with uniformization in algebraic/arithmetic geometry? E.g. this article by Faltings seems to be about that, the Shimura-Taniyama statement too, Mochizuki ...
- 10.8k
2
votes
4
answers
617
views
A question on function fields (extending my previous question)
Consider the extension $\mathbb Q(a,b)$ of the field of rationals, where $a$, $b$ are algebraically independent transcendentals. To $\mathbb Q(a,b)$ adjoin the roots of the polynomials $x^5+a^5=1$ and ...
- 161
15
votes
5
answers
3k
views
Can we count isogeny classes of abelian varieties?
Let's fix a finite field F and consider abelian varieties of dimension g over F. Can we say how many isogeny classes there are? Is it even clear that there's more than one isogeny class? For g=1, ...
- 1,988
6
votes
3
answers
2k
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 ...
- 191
2
votes
1
answer
354
views
k-th Chow Group and k-th graded part of K_0 ismorphic for DM-stacks?
If X is an algebraic scheme, K_0(X) has a filtration by taking the subgroups generated by coherent sheaves whose support as at most dimension k. The associated graded groups are the quotients, and ...
- 3,917
40
votes
5
answers
4k
views
How should one approach tropical mathematics?
Let me preface this by saying that my background is pretty meagre (i.e. solid undergrad). However, a few months ago I came across Litvinov - The Maslov dequantization, idempotent and tropical ...
98
votes
10
answers
14k
views
equivalence of Grothendieck-style versus Cech-style sheaf cohomology
Given a topological space $X$, we can define the sheaf cohomology of $X$ in
I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$)
or
II. the Čech ...
- 1,871
50
votes
1
answer
15k
views
Consequences of Geometric Langlands
So, lots of people work on the Geometric Langlands Conjecture, and there have been a few questions around here on it (admittedly, several of them mine). So here's another one, tagged community wiki ...
45
votes
6
answers
6k
views
Universal definition of tangent spaces (for schemes and manifolds)
Both schemes and manifolds are local ringed spaces which are locally isomorphic to spaces in some full subcategory of local ringed spaces (local models). Now, there is the inherent notion of the ...
- 5,233
18
votes
3
answers
5k
views
Why are local systems on a complex analytic space equivalent to vector bundles with flat connection?
Let $X$ be a complex analytic space. It is a 'well known fact' that the categories of local systems on $X$ (i.e. locally constant sheaves with stalk $C^n$), and of (holomorphic) vector bundles on $X$ ...
- 1,488
4
votes
1
answer
275
views
Comparing maps of reduced schemes
Nice fact:
Suppose f:X->Y is a map of schemes and Z⊆Y is a subscheme (locally closed immersion) containing the set-image of X. If X and Z are reduced, then it follows that f factors through Z. ...
- 11.2k
11
votes
1
answer
705
views
a question on function fields
Consider the transcendental extension Q(t) of the field of rationals.
To Q(t) adjoin the root of the polynomial x^5+t^5=1. The resulting
field Q(t)[x] is a radical extension of Q(t). Is it true that ...
- 161
4
votes
3
answers
715
views
Is a holomorphic vector bundle on a projective variety locally trivial in the Zariski topology?
By the GAGA principle we know that a holomorphic vector bundle E->X is analitically isomorphic to an algebraic one, say F->X, and by definition F is locally trivial in the Zariski topology. But since ...
- 14.7k
22
votes
2
answers
1k
views
Is Hodge theory somehow connected with a Galois group action Gal(C/R)?
I'm currently taking a course in Hodge theory ... and I wonder if all the splittings in $\{i,-i\}$ Eigenvalue pairs come from the Galois group action (of the extension $\mathbb{R}\rightarrow\mathbb{C}$...
- 1,926
56
votes
8
answers
8k
views
Questions about analogy between Spec Z and 3-manifolds
I'm not sure if the questions make sense:
Conc. primes as knots and Spec Z as 3-manifold - fits that to the Poincare conjecture? Topologists view 3-manifolds as Kirby-equivalence classes of framed ...
9
votes
3
answers
2k
views
Crepant resolutions of toric varieties
Given a toric variety, is it easy to see if a crepant resolution exists? If so, how can it be explicitly constructed?
- 611
3
votes
1
answer
1k
views
Iso-lines to 3D Surface Generation
I have a set of isolines points ( or contour points) such as this:
Each point has their own respective X, Y and Z. Since they are isolines, that means that all of the points will have a unique X-Y ...
- 381
17
votes
2
answers
1k
views
Can Hom_gp(G,H) fail to be representable for affine algebraic groups?
Let $G$ and $H$ be affine algebraic groups over a scheme $S$ of characteristic 0 and let $\textbf{Hom}_{S,gp}(G,H)$ be the functor $T \mapsto \text{Hom}\_{T,gp}(G,H)$
Theorem (SGA 3, expose XXIV, 7....
- 10.5k
16
votes
12
answers
11k
views
Are there any interesting connections between Game Theory and Algebraic Topology?
I've been learning game theory on my own and was just curious how it connected with previous things I've learned. So are there any interesting connections between Game Theory and Algebraic Topology? ...
- 809
2
votes
1
answer
167
views
Triviality of the Hodge bundle for a special family of semistable curves
Let g,h be positive integers. Let E be an elliptic curve, C be a genus h curve, and D be a genus g-h-1 curve. Let c,d,e be points on (resp.) C,D, and E.
Let f:CC --> E-e be the family whose fiber ...
- 10.5k
25
votes
4
answers
4k
views
What are the automorphism groups of (principally polarized) abelian varieties?
What are the possible automorphism groups of a principally polarized abelian variety $(A,\lambda)$ of dimension $g,$ say an abelian surface ($g=2$) over the complex numbers or algebraic closure of a ...
- 4,265
0
votes
1
answer
485
views
Understanding a lemma in "Loop Spaces and Langlands Parameters" article
First, some background. I was trying to read the article Loop Spaces and Langlands Parameters but I get immediately stuck at Theorem 2.1 in the introduction.
This was actually forward-referring to ...
- 15.1k
13
votes
1
answer
1k
views
When do six operations work?
This question comes (heavily edited) from my notes, thus slightly unusual structure.
We know that algebraic maps have very strict structure, and in many settings the operations ...
- 15.1k
11
votes
4
answers
3k
views
What does ramification have to do with separability?
Does ramification have anything to do with inseparability? It feels like an extension of Q in which p ramifies should somehow correspond to an extension of F_p(t). Does totally ramified <--> purely ...
- 15.4k
11
votes
3
answers
2k
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 ...
- 16k
15
votes
3
answers
2k
views
Polynomials that are sums of squares
Is any algorithm known for determining whether or not a multivariate polynomial with integer coefficients can be written as a sum of squares of such polynomials?
By way of background, if we one ...
- 151
13
votes
3
answers
1k
views
Decomposition of k[G]
There's a well-known decomposition of $L^2(G)$, a regular representation of compact complex group Lie $G$, called Peter-Weyl theorem.
Turns out for some reason I automatically think that there is a ...
- 15.1k
3
votes
2
answers
1k
views
Castelnuovo Positivity (Rewrite of: Weil's original proof for FP^2)
Weil's proof of the Riemann Hypothesis for projective curves relies upon the following positivity result: Let $\mathbb{F}q$ be the finite field with $q$ elements, $\overline{\mathbb{F}q}$ its closure, ...
- 1,462
15
votes
2
answers
2k
views
Total Spaces of Quasicoherent Sheaves
You can construct a total space of a quasicoherent sheaf on an scheme by taking relative spec of the symmetric algebra of the dual sheaf. For locally free sheaves, you get vector bundles, and every ...
- 5,548
16
votes
3
answers
3k
views
When does direct image with proper support have a right adjoint?
For $f: X → Y$ a morphism of schemes, does anybody know conditions for the existence of an adjunction $(f_!,f^!)$ between the module-categories (not the quasicoherent), where $f_!$ is direct image ...
- 12.3k
3
votes
2
answers
242
views
Vector spaces of singular planar cubics
What is the largest dimensional linear space of singular planar cubics? Is this known?
Think of the space of planar cubics as a PP^9 (parametrized by the coefficients). The discriminant \Delta is ...
- 2,955
2
votes
1
answer
173
views
Projective Curves which are Principal Bundles
I have a very specific question: does anyone know of a (non-trivial) example of a projective curve which is also a homogenous space (or just a principal bundle)? The trivial example being CP^1 = SU(2)/...
- 1,462
8
votes
3
answers
1k
views
Is there a stable algorithm for polynomial division (in several variables)?
Suppose you have a homogeneous ideal $I$ inside the algebra $\mathbb{C}[x_1,...,x_d]$ of complex polynomials in $d$-variables. Can one find a basis for $I$, say $\{f_1,...,f_k\}$, such that every $h \...
- 550
3
votes
1
answer
320
views
limits of algebraic varieties
I'm looking for a reference which deals with limits of families of algebraic varieties as the degree increases (or at least keywords from this subject).
For the kind of example I have in mind, ...
- 2,901
18
votes
4
answers
2k
views
What are the Benefits of Using Algebraic Spaces over Schemes?
I have heard that algebraic spaces have better formal properties than schemes. What are these benefits? Also, is there a natural way to go straight from affine schemes to algebraic spaces bypassing ...
- 5,548