# Questions tagged [ag.algebraic-geometry]

Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

7,026
questions with no upvoted or accepted answers

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(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_{\...

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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 ...

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When topologists speak of an "immersion", they are quite deliberately describing something that is not necessarily an "embedding." But I cannot think of any use of the word "embedding" in algebraic ...

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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 ...

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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 ...

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In the 1989, Bob Thomason left his CNRS position in Orsay and moved to Paris VII. It was during this period that he composed his "Open Letter" to the mathematical community. The letter ...

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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 ...

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When I started to learn mathematics, I was fascinating by legendary «Демидович»: problems in mathematical analysis. Fifteen years later, when I open chapters about integrals, I see a long list of ...

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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 ...

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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 ...

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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 ...

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It is easy to show, with an explicit construction, that a homogeneous cubic function $f: \mathbb{Z}^2 \to \mathbb{Z}$ is not injective. I am seeking a proof of the same result without the condition ...

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

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(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 ...

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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 ...

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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 ...

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Let $X$ be a relatively nice scheme or topological space.
In various physics papers I've come accross, the Todd class $\text{Td}(T_X)$ is viewed as the Euler class of the normal bundle to $X\to LX$. ...

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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 ...

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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 ...

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I was reading the Russian translation of Recoltes et Semailles and in the footnote where Grothendieck lists his 12 contributions (including schemes) we find the following lines:
Из этих тем ...

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All rings are commutative with unit. A ring $R$ is called regular if it satisfies
(Reg) Every finitely generated ideal of $R$ has finite projective dimension.
Clearly this gives the usual ...

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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 ...

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Suppose that:
$X$ is a smooth complex algebraic variety,
$f : X \to D$ is a proper map to a small disc, smooth away from 0,
$Z_\epsilon = f^{-1}(\epsilon)$, and $Z = Z_0$.
Then there is a procedure (...

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I believe that running the spectral sequence of a filtered complex / spectrum $ \cdots \to F_n \to F_{n+1} \to \cdots$ can be viewed as doing deformation theory in some very primitive "derived ...

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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 ...

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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 ...

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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-...

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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 ...

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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 ...

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Consider the following theorem of Atiyah.
Let $X$ be a connected smooth projective curve over an algebraically closed field $k$. Then a vector bundle $L$ on $X$ admits a connection if and only if the ...

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In this text, the author says (well, he says it in French, but I am too lazy to fix all the accents, so here is a Google translation):
In any case, contemporary mathematics provides an example of ...

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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 ...

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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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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 ...

22
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In their influential paper Noncommutative Smooth Spaces (https://arxiv.org/abs/math/9812158), Kontsevich and Rosenberg define the notion of a noncommutative projective space. In Section 3.3 they ...

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Two questions:
Q1. Have researchers worked out minimum-degree
real algebraic curves in $\mathbb{R}^3$ realizing specific knots?
Some work on the trefoil is reported in this MSE question.
&...

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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 ...

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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 ...

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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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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 ...

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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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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 $...

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Let $X$ be a smooth connected quasiprojective variety over $\mathbb{C}$ and let $G$ be a complex reductive group. Let $$\iota: G\to GL_N$$ be a representation and let $$\rho: \pi_1(X(\mathbb{C}))\to G(...

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Let $X$ be a (not necessarily smooth or proper) variety over a number field $k$. Suppose we are given
A subquotient $V_{dR}$ of the algebraic de Rham cohomology $H_{dR}^i(X)$ (defined in the non-...

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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 ...

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Let $X = [P/G]$ be a smooth finite type separated DM-stack over $\mathbb C$ given as the quotient of a smooth quasi-projective scheme $P$ by the action of a smooth (finite type separated) reductive ...

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Let $F$ be a number field, $S$ a finite set of places, and $X$ a smooth projective $\mathscr{O}_{F,S}$-scheme with geometrically connected fibers. For each point $t\in \text{Spec}(\mathscr{O}_{F,S})$, ...

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The affine Grassmannian of a complex reductive group $G$ (for simplicity one can assume $G=GL_n$) admits the structure of a complex topological space. More precisely, the functor $$X\mapsto |X^{an}|$$ ...

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Artin's representability theorem gives conditions for a functor from commutative rings to sets (or groupoids) to be representable by an algebraic space (stack). The conditions are largely expressed ...

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Let $(\Sigma_g, x)$ be a pointed topological surface of genus $g$, and let $MCG(g,1)$ be the mapping class group of this pointed surface. Then $MCG(g,1)$ has a natural action on $\pi_1(\Sigma_g, x)$ $$...