# Questions tagged [ag.algebraic-geometry]

for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

4,348 questions

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### Grothendieck-Teichmuller conjecture

(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmuller conjecture: the morphism
$$
G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})
$$
is an isomorphism.
Here $G_{\...

**70**

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10k views

### Hironaka's proof of resolution of singularities in positive characteristics

Recent publication of Hironaka seems to provoke extended discussions, like Atiyah's proof of almost complex structure of $S^6$ earlier...
Unlike Atiyah's paper, Hironaka's paper does not have a ...

**63**

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2k views

### Why do combinatorial abstractions of geometric objects behave so well?

This question is inspired by a talk of June Huh from the recent "Current Developments in Mathematics" conference: http://www.math.harvard.edu/cdm/.
Here are two examples of the kind of combinatorial ...

**42**

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963 views

### Enriched Categories: Ideals/Submodules and algebraic geometry

While working through Atiyah/MacDonald for my final exams I realized the following:
The category(poset) of ideals $I(A)$ of a commutative ring A is a closed symmetric monoidal category if endowed ...

**38**

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1k views

### What is Prismatic cohomology?

Prismatic cohomology is a new theory developed by Bhatt and Scholze; see, for instance, these course notes. For the sake of the community, it would be great if the following question is discussed in ...

**38**

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1k views

### Uniformization over finite fields?

The following is a question I've been asking people on and off for a few years, mostly out of idle curiosity, though I think it's pretty interesting. Since I've made more or less no progress, I ...

**34**

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907 views

### Computer calculations in A_infinity categories?

Is there a good computer program for doing calculations in A-infinity categories?
Explicit calculations in A-infinity categories are an important, useful, yet very tedious task. One has to keep track ...

**33**

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1k views

### Mikhalkin's tropical schemes versus Durov's tropical schemes

In Mikhalkin's unfinished draft book on tropical geometry, (available here) (page 26) he defines a notion of tropical schemes. It seems to me that this definition is not just a wholesale adaptation of ...

**32**

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2k views

### Microlocal geometry - A theorem of Verdier

(1) In "Geometrie Microlocale", Verdier states the following theorem.
Theorem: Let $E$ be a vector space and $F$ a constructible complex on $E$.
Then for $\ell$ a linear form on $E$, we have a ...

**31**

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641 views

### Is there a rigid analytic geometry proof of the functional equation for the Riemann zeta function?

The adèles $\mathbb A$ arise naturally when considering the Berkovich space $\mathcal M(\mathbb Z)$ of the integers. Namely, they are the stalk $\mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p$ ...

**31**

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1k views

### Grothendieck's “List of classes of structures”

In Lawvere's article Comments on the Development of Topos Theory, the author writes:
Similarly, Grothendieck and others unerringly recognized which kinds of mathematical structures are 'preserved ...

**31**

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1k views

### History of the Proj construction in algebraic geometry

Projective geometry was introduced by fifteenth century Renaissance painters (like Alberti, da Vinci and Dürer) in the guise of perspective theory, although one could argue that Pappus was already ...

**28**

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2k views

### Is there software to compute the cohomology of an affine variety?

I have some affine varieties whose cohomology (topological, with $\mathbb{C}$ coefficients) I would like to know. They are very nice, they are all of the form $\mathbb{A}^n \setminus \{ f=0 \}$ for ...

**27**

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960 views

### Enriques surfaces over $\mathbb Z$

Does there exist a smooth proper morphism $E \to \operatorname{Spec} \mathbb Z$ whose fibers are Enriques surfaces?
By a theorem of, independently, Fontaine and Abrashkin, combined with the Enriques-...

**27**

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614 views

### Why do H_4 and M_4 have the same virtual Euler characteristic?

Here's a funny coincidence:
The virtual (or "orbifold") Euler characteristic of $\mathcal M_g$ is known by the work of Harer and Zagier: one has $\chi(\mathcal M_g) = \zeta(1-2g)/(2-2g)$.
Now ...

**26**

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1k views

### derived category of equivariant coherent sheaves and fixed points

The K-group $K^T(X)$ of $T$(torus)-equivariant coherent sheaves on a variety $X$ is isomorphic to $K^T(X^T)$, that of the fixed point locus via the inclusion homomorphism, when we tensor the quotient ...

**26**

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1k views

### Is there a Grothendieck-Riemann-Roch type of theorem generalizing Grothendieck's Lefschetz trace formula

Grothendieck deduced that the L-function of a (constructible) $\ell$-adic sheaf on a variety over $\mathbf{F}_p$ is rational from the generalized trace formula.
My first question is based on the ...

**26**

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2k views

### What are the possible singular fibers of an elliptic fibration over a higher dimensional base?

An elliptic fibration is a proper morphism $Y\rightarrow B$ between varieties such that the fiber over a general point of the base $B$ is a smooth curve of genus one.
It is often required for the ...

**25**

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513 views

### Mathieu group $M_{23}$ as an algebraic group via additive polynomials

An elegant description of the Mathieu group $M_{23}$ is the following: Let $C$ be the multiplicative subgroup of order $23$ in the field $F=\mathbb F_{2^{11}}$ with $2^{11}$ elements. Then $M_{23}$ is ...

**25**

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948 views

### Status of the Euler characteristic in characteristic p

In the introduction to the Asterisque 82-83 volume on `Caractérisque d'Euler-Poincaré, Verdier writes:
Enfin signalons que la situation en caractéristique positive est loin
d'être aussi ...

**24**

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601 views

### A modern perspective on the relationship between Drinfeld modules and shtukas

Shtukas were defined by Drinfeld as a generalization of Drinfeld modules. While the relationship between the definitions of Drinfeld modules and shtukas is not obvious, one does have a natural ...

**24**

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1k views

### The most important facts, modern surveys, and readable introductions to p-adic cohomology theories (crystalline cohomology and the mysterious functor)

I would like to organize a seminar on crystalline cohomology; I dream of understanding the Beilinson's recent paper on the mysterious functor (http://www.ams.org/journals/jams/2012-25-03/S0894-0347-...

**23**

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364 views

### Topological loops vs. algebro-geometric suspension in Hochschild homology

Let $k$ be a base commutative ring, and let $A$ be a (unital but not necessarily commutative) $k$-algebra. The cone on $A$ is the ring $CA$ of infinite matrices $(a_{ij})_{i,j \geq 1}$ that are ...

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2k views

### What's the intuition between formal smoothness, etaleness. and unramifiedness?

Let $f: X \to Y$ be a morphism of schemes. Then $f$ is called (EGA IV.17) formally smooth if whenever $T$ is an affine $Y$-scheme and $T'$ a closed subscheme of $T$ defined by a nilpotent ideal (it's ...

**22**

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661 views

### bound on the genus of a fiber of the Albanese map of a surface with $h^1({\mathcal O})=1$?

This is maybe more an open problem than a question, since I have seriously thought about it and asked several people working on algebraic surfaces with no success. I hope somebody here can ...

**21**

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497 views

### What is a Green's function in the language of $\mathcal{D}$-modules?

Let $P$ be an analytic linear differential operator defined on some open interval $X=(a,b)$ and $\mathcal{M}=\mathcal{D}_X / \mathcal{D}_X \bullet P$ the corresponding $\mathcal{D}$-module. I'm trying ...

**21**

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784 views

### What is classified by the (big) crystalline topos?

In his paper "Generic Galois Theory of Local Rings", G.C. Wraith states on p. 743 that the (big) crystalline topos "can be conveniently described in terms of the theory it describes". What exactly is ...

**20**

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505 views

### Nonabelian topological fundamental group of a conjugate variety

Let $X$ be a pointed algebraic variety over the field of complex numbers $\mathbb{C}$.
Let $\pi_1^{\rm top}(X)$ and $\pi_1^{\mathrm{\acute{e}t}}(X)$ denote the topological and the étale fundamental ...

**20**

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495 views

### Bounding failures of the integral Hodge and Tate conjectures

It is well know that the integral versions of the Hodge and Tate conjectures can fail. I once heard an off hand comment however that they should only fail by a "bounded amount". My question is what ...

**19**

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696 views

### Finiteness of etale cohomology for arithmetic schemes

By an arithmetic scheme I mean a finite type flat regular integral scheme over $\mathrm{Spec} \, \mathbb{Z}$.
Let $X$ be an arithmetic scheme. Then is $H_{et}^2(X,\mathbb{Z}/n\mathbb{Z})$ finite ...

**19**

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440 views

### Can we just use effective descent morphisms (pure morphisms) as covers?

There are a number of notions of "cover" for a scheme: etale, faithfully flat, fpqc, fppf, Zariski, Nisnevich, etc. Most of these have a nice property, which is that a cover of that type satisfies ...

**19**

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842 views

### Is there a functor of points approach to algebraic cycles and intersection theory?

Motivation
Most of the algebraic geometry I have done so far was concerned with group schemes (e.g., abelian schemes, tori, unipotent groups). In that part of the field the "functor of points POV" is ...

**18**

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485 views

### Is every $p$-group the $\mathbb{F}_p$-points of a unipotent group

Let $\Gamma$ be a finite group of order $p^n$. Is there necessarily a unipotent algebraic group $G$ of dimension $n$, defined over $\mathbb{F}_p$, with $\Gamma \cong G(\mathbb{F}_p)$?
I have no real ...

**18**

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497 views

### Has the following problem posed by Deligne in the official description of the Hodge conjecture been solved?

Towards the end of his official description of the Hodge conjecture, Deligne asked the following question:
Let $A$ be an abelian variety over the algebraic closure $\mathbb{F}$
of a finite field $...

**18**

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429 views

### Does the “holomorphic spheres-to-continuous spheres” forgetful function respect the mixed Hodge structures on homotopy groups?

For each smooth, projective, complex variety $X$ that is simply connected, John Morgan constructed a natural mixed Hodge structure on the homotopy group $\pi_k(X,x)\otimes \mathbb{Q}$. This was ...

**18**

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447 views

### What should motives for $L(E,n)$ look like?

Goncharov and Manin showed in this paper that the zeta values $\zeta(n)$ can be realized as periods of framed mixed Tate motives constructed from moduli spaces $\overline{\mathcal{M}}_{0,n+3}$ of ...

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950 views

### Hrushovski's Construction

Zilber expressed a conjecture for $\aleph_{1}$- categorical theories (In the 80s).
Zilber's Conjecture: The geometry of any $\aleph_{1}$- categorical structure is one of the following:
(a) Trivial (...

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926 views

### Schemification (schematization?) of locally ringed spaces

Motivation:
Say $F: D \to Sch$ is a diagram in the category of schemes, and we're interested in whether it has a colimit (gluings, pushouts, and "categorical" quotients are all examples of colimits)....

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### Two conjectures by Gabber on Brauer and Picard groups

In a paper I need to make reference to 2 conjectures by Gabber
(see Conjectures 2 and 3, page 1975)
http://www.mfo.de/programme/schedule/2004/32/OWR_2004_37.pdf
1) Let $R$ be a strictly henselian ...

**17**

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446 views

### Which rings are cohomology rings?

Which rings can arise as cohomology rings of algebraic varieties?
To be more specific, take a Weil cohomology theory $H^*$ with coefficients in a field $K$ of characteristic 0 defined for smooth ...

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328 views

### Is there a useful theory of D-modules on smooth (non-analytic) manifolds?

D-modules are related to flat connections on vector bundles, end hence local systems. The theory of D-modules (and related notions such as crystals etc.) seems to be very popular in complex analytic ...

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950 views

### History of the functor of points

Until now, I thought the functor of points approach was introduced by Grothendieck at the 1973 Buffalo seminar.
However, in this note by Lawvere the author writes:
"I myself had learned the ...

**17**

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508 views

### Smooth curves on smooth varieties

Let $X$ be a smooth, proper algebraic variety over a field $k$, of positive dimension.
Is it true that $X$ contains a smooth Zariski-closed curve?
If it is projective, this is true by Bertini. But ...

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561 views

### Are there lightweight foundations for arbitrarily extendable objects?

My experience with foundations is rather scant, but I've run into some types of objects that seem to resist the sort of set-theoretic encoding schemes via Kurowski tuples that are rather common for ...

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### Idea of presheaf cohomology vs. sheaf cohomology

Let $X$ be a topological space and $U$ an open cover of $X$.
In this thread Angelo explained beautifully how presheaf cohomology (Cech cohomology) relates to sheaf cohomology:
The zeroth Cech ...

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### Vector Bundles on the Moduli Stack of Elliptic Curves

As is well known, there is classification of line bundles on the moduli stack of elliptic curves over a nearly arbitrary base scheme in the paper The Picard group of $M_{1,1}$ by Fulton and Olsson: ...

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426 views

### What are hyperkähler metrics used for?

It seems that a lot of effort has been devoted to endow holomorphic-symplectic manifolds with hyperkähler metrics. It started with Calabi [4] with $T^*\mathbb{CP}^n$. Other examples include coadjoint ...

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406 views

### Are there any examples of hyperbolic curves over finite fields such that the action of frobenius on its prime-to-$p$ fundamental group is known?

Let $X$ smooth curve over a finite field $\mathbb{F}_q$ of type $(g,n)$ - that is, $X$ is an open subscheme of its genus $g$ compactification obtained by removing $n$ points.
Any such curve ...

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378 views

### Is there a cotangent bundle of a stable $\infty$-category?

Let $C$ be a stable $\infty$-category. Is there any categorical construction $C \mapsto T^* C$, where $T^* C$ is another stable $\infty$-category, that specializes to the following?
When $C$ is the ...

**16**

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664 views

### What is the expected dimension of the Zariski closure of the rational points on the moduli space of curves?

For each genus $g$, there are many curves of genus $g$ defined over $\mathbb Q$. How many? We might study this question by considering the rational points of the Deligne-Mumford moduli space of curves ...