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Questions tagged [algebraic-geometry]

The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. This tag should not be used for elementary problems which involve both algebra and geometry.

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Pullback and Pushforward Isomorphism of Sheaves

Suppose we have two schemes $X, Y$ and a map $f\colon X\to Y$. Then we know that $\operatorname{Hom}_X(f^*\mathcal{G}, \mathcal{F})\simeq \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$, where $\...
Matt's user avatar
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108 votes
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Classification of local Artin (commutative) rings which are finite over an algebraically closed field

A result in deformation theory states that if every morphism $Y=\operatorname{Spec}(\mathcal{A})\rightarrow X$ where $\mathcal A$ is a local Artin ring finite over $k$ can be extended to every $Y'\...
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30 votes
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Proof of residue theorem (residue formula) for differential forms on curves over an arbitrary closed field.

I have been reading the book Algebraic Geometric Codes: Basic Notions by Tsfasman, Vladut and Nogin. They give a residue formula like this (Proposition 2.2.20, p. 97): Let $\mathbb{k}$ be an ...
Ashu Pachauri's user avatar
29 votes
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1k views

Recent developments in the proof of fermat's last theorem

It's been 20 years since fermat's last theorem was proved by Andrew Wiles. Has there been any simplification in proof in the last 20 years? What I do only know is that different proofs of faltings's ...
user779120's user avatar
25 votes
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775 views

When is a polynomial contained in the ideal generated by its partial derivatives?

Let $R = k[x_1,\dots,x_n]$ be a multivariate polynomial ring over a field $k$ of characteristic zero, and let $f\in R$. Is there an easy-to-test necessary and sufficient condition on $f$ such that $f$...
Ben Blum-Smith's user avatar
23 votes
1 answer
543 views

Prop. 3.2.15 in Liu: Geometrically reduced algebraic variety

I have a problem with proposition 3.2.15 of Algebraic geometry and arithmetic curves of Qing Liu: Let $X$ be an integral algebraic variety over $k$. If $X$ is geometrically reduced then $K(X)$ is a ...
Gabriel Soranzo's user avatar
23 votes
0 answers
568 views

When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ for $\Gamma$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least ...
simplequestions's user avatar
19 votes
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5k views

When is the direct image functor exact?

Consider a morphism of topological spaces $f:X\to Y$. The direct image functor takes a sheaf $\mathcal{F}$ on $X$ to the sheaf defined by $f_*\mathcal{F}(U)=\mathcal{F}(f^{-1}(U))$. It's a right ...
KReiser's user avatar
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19 votes
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Quasicoherent sheaves as smallest abelian category containing locally free sheaves

On page 362 of Ravi Vakil's notes, the author says "It turns out that the main obstruction to vector bundles to be an abelian category is the failure of cokernels of maps of locally free sheaves - ...
Arrow's user avatar
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19 votes
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(Weil divisors : Cartier divisors) = (p-Cycles : ? )

Suppose $X$ verifies the suitable conditions in which Weil (resp. Cartier) divisors make sense. The group of Weil divisors $\mathrm{Div}(X)$ on a scheme $X$ is the free abelian group generated by ...
Simplicius's user avatar
17 votes
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969 views

Algebraically closed fields of characteristic $0$ and $\mathbb{C}$

Let $k$ be an algebraically closed field of characteristic $0$. Then I have heard that if $k$ has cardinality no greater than that of $\mathbb{C}$, then there is an embedding $k\hookrightarrow\mathbb{...
oxeimon's user avatar
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16 votes
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Applications of $Ext^n$ in algebraic geometry

I have been doing a project about $\operatorname{Ext}^n$ functors for my commutative algebra class. I used the approach via extensions of degree n. Basically I have shown the long exact sequence ...
Abellan's user avatar
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16 votes
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Strong Nullstellensatz without the Rabinowitsch trick

All the textbooks I've seen prove the Strong Nullstellensatz from the Weak Nullstellensatz using the Rabinowitsch trick. How did they prove the Strong Nullstellensatz before the Rabinowitsch trick ...
AnatolyVorobey's user avatar
15 votes
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331 views

Algebraic methods to compute the cohomology ring of the complex topology of a variety?

Suppose $V$ is an affine (resp. projective) subvariety of the affine (resp. projective) space $\mathbb A_\mathbb C^n$ (resp. $\mathbb P_\mathbb C^n$) with vanishing ideal $I\subseteq\mathbb C[X_1,\...
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15 votes
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Weighted blow-ups

I would like to understand what's a weighted blow-up in a very simple case: $\mathbb{C}^2$ blown-up in the origin with weights $(a,b)$. In found some notes online saying that this is the surface $X$ ...
John Simis's user avatar
15 votes
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Why is the sheaf $\mathcal{O}_X(n)$ called the "twisting sheaf" (where $X=\operatorname{Proj}(S)$ for a graded ring $S$)?

Basically my question is why the sheaf $\mathcal{O}_X(n)$ is called the twisting sheaf, here $X=\operatorname{Proj}(S)$ and $S$ any graded ring.
sti9111's user avatar
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14 votes
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The Hasse invariant at cuspidal elliptic curves in a generically ordinary family

For an elliptic curve of the form $y^2 = f(x)$ where $f(x) \in \mathbb F_q[x]$ is a cubic polynomial with distinct roots, it is known (from Silverman's book, say) that the curve is supersingular ...
Arkady's user avatar
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14 votes
1 answer
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Can every variety appear as singular locus?

let $V$ be a projective variety. Does there always exist another variety $X$ such that $\operatorname{Sing}X=V$? Here $\operatorname{Sing}X$ means the singular locus of $X$ with reduced structure.
Saberization's user avatar
14 votes
0 answers
793 views

Maximum number of intersection points of two different Bernoulli lemniscates

What is the maximum number of intersection points of two different Bernoulli lemniscates in the real plane? (Of course two identical lemniscates share an infinite number of points.) Here are some of ...
Mostafa's user avatar
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14 votes
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479 views

A characterization of Henselian rings

It is well known that if $(A, \mathfrak m)$ is a Henselian local ring with residue field $\kappa$, then base-change from $A$ to $\kappa$ determines an equivalence of categories $$F: \{\text{Finite ...
Bruno Joyal's user avatar
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14 votes
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Noncommutative algebraic geometry in the case of skew fields

Noncommutative algebraic geometry is a developing field. Things have not yet got the final form as in commutative geometry. But one might wonder whether things are any better in the case of skew-...
user avatar
14 votes
1 answer
2k views

Finite morphism that is not projective

There are two definitions of a projective morphism. Hartshorne: A morphism $f: X\to Y$ is projective if it factors as $f=gi$, where $i: X\to P_Y^n$ is closed imbedding and $g: P_Y^n\to Y$ is ...
minimax's user avatar
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13 votes
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318 views

On the properties of sum-of-squares polynomials

Definition 1. If a multivariate polynomial $f$ can be written as a finite sum of squared polynomials, i.e., $f(x)=\sum_{i = 1}^n g_i^2(x)$, then $f$ is SOS. Definition 2. If an $n$-variate polynomial ...
khashayar's user avatar
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13 votes
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321 views

The étale topos of a scheme is the classifying topos of...?

By a theorem of Joyal and Tierney, every Grothendieck topos is the classifying topos of a localic groupoid. It has been proved (e.g. C. Butz. and I. Moerdijk. Representing topoi by topological ...
W. Rether's user avatar
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13 votes
0 answers
390 views

Applications of resolution of singularities

I would like to know applications of Resolution of Singularities. What are the benefit of resolving singularities of a variety by blow-up maps in a context outside of mathematics? I'm fine with both ...
AmirHosein Sadeghimanesh's user avatar
13 votes
0 answers
355 views

Motivation?: Lie algebra and algebraic group Cohomology

This is just an apriori question to get a motivational heuristic idea: If an algebraic group G (more generally, G an affine group scheme), is connected over an algebraically closed base-field k. ...
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In which commutative algebras does any derivation possess a flow?

Suppose $A$ is a commutative algebra over $\mathbb{R}$ with unity. $\mathbb{R}$-linear map $\xi\colon A\to A$ is a derivation of $A$ iff $\xi(ab)=a\xi(b)+\xi(a)b$ for any $a,b\in A$. If $\gamma\colon \...
Fiktor's user avatar
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13 votes
1 answer
386 views

Does every vector bundle have a 'tensor inverse'?

For any vector bundle $E$ over a finite-dimensional CW complex, there is a vector bundle $E'$ such that $E\oplus E'$ is trivial. For a compact Hausdorff base, this is Proposition 1.4 of Hatcher's ...
Michael Albanese's user avatar
12 votes
0 answers
215 views

Local systems defined by higher homotopy groups

I should mention I have very little background in algebraic topology and don't really know much about homotopy groups besides the definition. I am aware that for a topological space $X$ and a point $x ...
Thomas Manopulo's user avatar
12 votes
0 answers
1k views

Morphisms with connected fibers

Let $f\colon X\to Y$ be a morphism of schemes. I am interested in the following property (too long for the title): $$ f_{*}\mathcal{O}_{X}=\mathcal{O}_{Y} \quad \text{(P)}$$ Under very reasonable ...
Pedro's user avatar
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12 votes
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Understanding pullback of pushforward

Sorry if this is sort of a soft question. I added an example at the end to mitigate. So I hope you can bear with me here. Let $\phi : X \rightarrow Y$ be a morphism of noetherian schemes, $\mathscr F$...
rollover's user avatar
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12 votes
0 answers
444 views

Fiber product of toric varieties

Let $X$,$Y$ and $Z$ be three toric varieties defined by the fans $\Sigma_X\subset (N_X)_{\mathbb{R}}$, $\Sigma_Y\subset (N_Y)_{\mathbb{R}}$ and $\Sigma_Z\subset (N_Z)_{\mathbb{R}}$, respectively. It ...
Pedro Montero's user avatar
11 votes
0 answers
166 views

A natural topology on a field

I can endow any field with a natural topology in the following way. Given a polynomial $f\in K[X]$, I denote by $\mathcal{O}(f)=\{x\in K\mid\exists y\in K^{\times}\ f(x)=y^2\}$, i.e. the set of ...
Jacques's user avatar
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11 votes
0 answers
325 views

Is a complex algebraic set with a Zariski dense subset of algebraic points already defined over the algebraic numbers?

Edit: I now crossposted this question on MO: https://mathoverflow.net/questions/428384/is-a-complex-algebraic-set-with-a-zariski-dense-subset-of-algebraic-points-alrea Let $X$ be a complex algebraic ...
Luvath's user avatar
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11 votes
0 answers
213 views

Example where the associated sheaf does not exist

What is a simple example of a topological space $X$ and a complete category $\mathcal{C}$ such that the inclusion $\mathbf{Sh}(X,\mathcal{C}) \hookrightarrow \mathbf{PSh}(X,\mathcal{C})$ has no left ...
Martin Brandenburg's user avatar
11 votes
0 answers
189 views

Algebraic Geometric Analogue of Brown's Representability

Brown's representability theorem is very usefull to show that the functor $$X \rightarrow H^i(X,A)$$ is representable. I would be interested to see if there exists an analogue of this statement in ...
curious math guy's user avatar
11 votes
0 answers
346 views

G-equivariant isomorphism inducing isomorphisms on quotients

Suppose that $X$ and $Y$ are smooth quasi-projective varieties over $\mathbb{C}$ with a holomorphic map $f:X \to Y$ inducing isomorphisms $f_* : H_i(X;\mathbb{Q}) \to H_i(Y;\mathbb{Q})$ for all $i \...
jacob's user avatar
  • 193
11 votes
0 answers
669 views

Null-correlation and Tango bundles on $\mathbb{P}^3$

Let $V$ be a four-dimensional complex vector space and $\mathbb{P}^3=\mathbb{P}(V)$. There are two interesting bundles $N$ and $T$ on $\mathbb{P}^3$, both of rank 2, called respectively a null-...
mahavishnu's user avatar
11 votes
0 answers
5k views

How a blow up changes the Canonical bundle?

lt $f:Y\longrightarrow X$ be a blowup a subvariety $V\subset$, where say both X and Y are smooth. Then what is the relation of $K_Y$ and $K_X$ ? The case of surfaces is clear What about higher ...
Shuhang's user avatar
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11 votes
0 answers
324 views

Galois cohomologies of an elliptic curve

I am studying basic theory of elliptic curves. I encountered Galois cohomology. But two introductory textbooks I read used only $H^0$ and $H^1$. I am curious why higher cohomologies did not appear. I ...
user avatar
11 votes
0 answers
340 views

Finding irreducible components of Spec$(R/I^n)$

Let $R= k[x,y,z]/(xy,yz,zx)$. Let $I=(x)$. What are the irreducible components of $\mathrm{Spec}(R/I^n)$ where $n \geq 2$ and $k$ is a field? For solving this problem I'm trying to use following ...
Arpit Kansal's user avatar
  • 10.3k
11 votes
0 answers
499 views

Singularities on a weighted projective curve

Let $C$ be the curve of degree $3$ defined over $\mathbb{C}$ given by $$x(y+z)=y^3-z^3$$ which lives in the weighed projective space $\mathbb{P}(x,y,z)=\mathbb{P}(2,1,1)$. Is the curve singular ?
Mr. No's user avatar
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11 votes
0 answers
505 views

27 lines on a smooth cubic surface

It is known that every smooth cubic surface with coefficients in $\mathbb{Q}$ has $27$ lines defined over a number field extension of $\mathbb{Q}$ of degree at most $51840$ as the group $\operatorname{...
Mr. No's user avatar
  • 1,061
11 votes
0 answers
377 views

Are the concepts of scheme-theoretic complete intersection and Ideal theoretic complete intersection the same?

Suppose $X$ is a projective variety in $\Bbb{P}^n$ of dimension $k$. Everything is over a field. We say $X$ is a scheme-theoretic complete intersection if $X$ can be written as $V_+(f_1) \cap \ldots \...
user avatar
11 votes
0 answers
705 views

Global sections of vector bundles on a complex elliptic curve and analytic functions

Let me fix an elliptic curve $E$ over complex numbers with distinguished point $x \in E$. Thanks to Atiyah we know everything about discreet parameters of vector bundles and its moduli spaces. But I ...
Alex's user avatar
  • 6,284
11 votes
0 answers
562 views

algebraic $1$-forms vs analytic $1$-forms

First let's fix some definitions: Definitions: Complex manifold (of dimension n): Is a locally ringed space $(X,\mathscr F)$, where there is an open cover $\bigcup_{i\in I} U_i=X$ such that $(U_i,\...
Dubious's user avatar
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11 votes
0 answers
2k views

Gluing schemes Hartshorne example

In example 2.3.5 Hartshorne says Let $X_1$ and $X_2$ be schemes. Let $U_1 \subseteq X_1$ and $U_2 \subseteq X_2$ be open subsets, and let $\varphi: ( U_1, \mathcal{O}_{X_1 \mid U_1}) \to ( U_2, \...
Grobber's user avatar
  • 3,258
11 votes
0 answers
413 views

Transformations that map points inside the sphere to points inside the sphere

I am trying to figure out what is the most general linear transformation that maps points inside the unit sphere to points inside the unit sphere. I am slightly abusing the word linear here by ...
Nicolás Quesada's user avatar
11 votes
0 answers
1k views

Self-Intersection Number $-2$

I am new here, but hopefully you can help me with a concrete problem I have. I try to compute a Self-Intersection Number of a constructed curve in an analytic surface. I know the answer by some ...
Greyfox's user avatar
  • 263
11 votes
1 answer
3k views

Global sections of Proj

In constructing the relative Proj for a graded ring $S$, one inevitably has that the distinguished opens, $D_+ (f)$, where $f \in S_+$ gives rise to a scheme that is isomorphic to $\textrm{Spec } S_{(...
Future's user avatar
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