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.
31,075 questions
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Chern class of tangent bundle of hypersurface of CP^n viewed as complex of coherent sheaves
I would like to compute the total Chern class of the tangent bundle to a hypersurface $X$ of degree $d$ in $\mathbb{P}^n$ by viewing the following short exact sequence as a complex of coherent sheaves ...
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Construction of Base Change/Comparison Map $g^*f_*F \to f'_* g'^*F$
Let us consider a cartesian diagram of schemes
$$
\require{AMScd}
\begin{CD}
X'=X \times_S S' @>{g'} >> X \\
@VVf'V @VVfV \\
Y' @>{g}>> Y
\end{CD}
$$
and let $F$ a sheaf on $X$.
...
- 10.1k
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66
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Quotient of $\mathrm{GL}_2(\mathbb{C})$ by a finite group
Let us consider the algebraic group $G=\mathrm{GL}_2(\mathbb{C})$ and consider the $S_2$-action given by conjugation with $P_0=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, that is, the $S_2$-...
- 181
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When do coequalizers of algebraic stacks exist?
This question might need some work to actually get a "good" answer.
Here's the background motivation: the $2$-category of algebraic stacks has fibre products and products and so has ...
- 427
3
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Why elliptic curves instead of Edwards curves?
I have been wondering this for a while; I get how elliptic curves of the Weierstrass form $y^2=4x^3-g_2x-g_3$ have the lowest degree and are the simplest way to study a genus $1$ curve. But why aren’t ...
- 360
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53
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Preimage of curves in a double cover of surface
Let $S$ be a smooth complex projective surface. We consider the double cover $\pi:X\to S$ branched along $B\equiv 2L$(suppose $B$ is smooth). Now let $C\subset X$ be an irreducible curve.
If $C$ is ...
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Are there any lines on an Abelian surface?
I'm working in the category of schemes over $\mathbb C$ or algebraic varieties over the same. Here by line I mean any curve isomorphic to $\mathbb P^1$, of any degree; i.e. the twisted cubic is a line....
- 199
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When is an open immersion affine as a morphism of schemes?
This might be a stupid question but I could not convince myself of the answer.
Let $j:X \to Y$ be an open immersion of schemes, and assume that $j$ is affine. Very broadly, what can we say? More ...
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Hironaka’s polyhedron.
Hironaka proved the resolution of singularities for varieties over characteristic zero. He invented his original invariant associated to the given singular loci. I remember that after blowing ups, his ...
- 1,067
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Quot scheme of a line on $\Bbb A^3$
Let $L \subset \Bbb A^3$ be a line. I'm trying to compute the dimension of the scheme $\operatorname{Quot}_{\Bbb A^3}(\mathscr{I}_L,2)$ and show that it is singular, but I have a bit of trouble with ...
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When is a zero-dimensional subscheme general or special?
On complex algebraic surfaces, say $X$, I'm working on some cohomology calculations of sheaves of ideals of the form $\mathcal{I}_{Z}(L)$, where $L\in\text{Pic}(X)$ and $Z\subset X$ is a zero-...
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Reduced structure commutes with base change
Let $X$ be a scheme, and let $K$ be a characteristic $0$ field.
This post shows that if $X$ is reduced, then base change over $X$ is also reduced. Is it also true that
$$X_{\mathrm{red}}\times_{\...
- 1,656
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how to show that a family of affine curves is non-singular for a family which produces infinitely many curves [closed]
I am working on a proof, and my strategy is to basically prove that this infinite family has no singularity within itself, and then using Siegel I could just prove that there are finitely many ...
- 11
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An algebraic structure with uniquely complemented and non-distributive
Is there a concrete example of an algebraic structure with two binary operations that is commutative, associative, uniquely complemented, and non-distributive?
Here is an explicit example that is easy ...
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164
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A Consequence from Grauert's Result on Cohomology of Fibres
Let $f:X \to Y$ be a map of schemes and $y \in Y$ a point with residue field $\kappa(y)$. Let $\mathcal{F}$ be a coherent sheaf on $X$. Then there exist a comparison map
$$ f_*(\mathcal{F}) \otimes \...
- 10.1k
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Solutions of a non-linear system
Consider a system of equations $$dd_i=\frac{f}{c}\boldsymbol{v}\cdot\left(\frac{\boldsymbol T}{\|\boldsymbol T\|}-\frac{\boldsymbol T-\boldsymbol p_i}{\|\boldsymbol T-\boldsymbol p_i\|}\right),$$
...
- 937
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Scheme over base field $k$ which is algebraically (resp. separably) closed in ring of regular functions of $X$
Let $X$ be a scheme over base field $K$ and denote by $R[X](= \Gamma(X,\mathcal{O}_X))$ the ring of regular global functions on $X$.
Question: Assume that $k$ is algebraically (resp. separably) closed ...
- 10.1k
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$\mathbb{C}(\lambda x^ay^b,\mu x^cy^d)=\mathbb{C}(x,y)$ if and only if $ad-bc= \pm1$
Let $a,b,c,d \in \mathbb{N}$ such that $ad-bc= \pm1$.
If I am not wrong, for such $a,b,c,d$ we have:
$\mathbb{C}(\lambda x^ay^b,\mu x^cy^d)=\mathbb{C}(x,y)$, where $\lambda,\mu \in \mathbb{C}-\{0\}$.
...
- 7,257
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1
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A seeming gap of chapters on local parameters and Taylor series in Shafarevich's algebraic geometry book
I seem to find a gap in Shafarevich's algebraic geometry book, located in Section $2$ of Chapter $2$, which is about local parameters and power series expansion. In the following paragraph I briefly ...
- 1,239
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Certifying that a system of polynomial equations has finitely many solutions
I have $n$ quadratic equations in $n$ variables. I can give $2^n$ different solutions. According to Bézout's theorem, if the system of equations has finitely many solutions, then these are all the ...
- 58.1k
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Does separated/proper model always exist [closed]
Let $X$ be a separated/proper scheme of finite type over a number field $k$. By spreading out, we know there is such one on $\mathcal O_{k,S}$. Do we know that there exists a separated/proper model ...
- 1,656
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Simplicial circle in motivic homotopy theory
I came up with a vauge question while reading this post. Does the simplicial circle $S^1$ have anything to do with the topology of $\mathbb{C}$-points of some variety? The linked question suggests ...
- 1,115
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Proof of embeddibility of projective smooth $k$-scheme with dimension $d$ in $\mathbb{P}^{2d+1}_k$ ( Part 2, Gortz, Wedhorn )
I am reading the Gortz, Wedhorn's Algebraic Geometry, proof on Theorem 14.132 and stuck at some statements. ( Can anyone who have the Gortz, Wedhorn's book help? )
EDIT : This post is not duplicate. ...
- 4,094
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1
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Conversion between Cartier Divisors, Weil Divisors, Line Bundles and Invertible Sheaves.
I am superficially quoting the following result from Algebraic Geometry by Hartshorne. I was told the following from a brief conversion with a graduate student, and became very interested. Please be ...
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Ideal of maximal minors is radical when rank never drops by more than 1 [Reference request]
Let $R := \mathbb C[x_1,\dots,x_d]/J$ be an affine domain which is the coordinate ring of the affine variety $X = V(J) \subseteq \mathbb C^d$. Let $M \in R^{m\times(n+1)}$ be a matrix with entries ...
- 2,908
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The nature of elements in local rings and local parameters
I have problems understanding the nature of elements in local rings and local parameters. Let $P$ be a point on a (quasiprojective) variety $X$.
I do know that
an element $f$ in the local ring $\...
- 1,239
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What makes the category of isocrystals over non-algebraically closed field non-semisimple?
In the following, I adopt the conventions in Rapoport & Zink's Period Spaces for $p$-divisible Groups.
Let $L$ be a perfect field of characteristic $p$. Let $W(L)$ be the ring of Witt vectors and ...
- 360
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Is it true that $s\in mM$ and $t\in mN$ implies $s\otimes t\in m(M\otimes N)$?
Let $M,N$ be both $A$-modules where $A$ is a commutative ring and let $m\subset A$ be a maximal ideal, is it true that $s\in mM$ and $t\in mN$ implies $s\otimes t\in m(M\otimes N)$?
This question is ...
- 305
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Let $k$ be an infinite field. Then $k$-valued points $V(k)$ of a nonempty open subscheme $V$ of a projective space $\mathbb{P}^{N}_k$ is nonempty?
Literally, let $k$ be an infinite field and $V$ a nonempty open subsheme of a projective space $\mathbb{P}_k^{N}$. Then the $k$-valued points $V(k)$ is nonempty? If so, how can we show? In particular ...
- 4,094
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Is the proj sheafification of graded canonical module the canonical bundle of smooth projective variety with Cohen-Macaulay coordinate ring?
Let $S=\oplus_{n\in \mathbb N} S_n$ be a positively graded ring such that $S_0=\mathbb C$ and $S$ is a finitely generated $\mathbb C$-algebra (I'm not assuming $S$ is generated in degree $1$). Assume $...
- 1,901
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projection formula and asymptotic Riemann-Roch
Let $X$ be a smooth projective variety of dimension $n$. Let $A$ be a nef divisor on $X$ and $Y$ be a dimension $n-1$ closed subscheme of $X$. Let $i: Y \to X$ be the inclusion map.
Let $N$ be a ...
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On hypersurface singularity.
Suppose that $A \colon= {\Bbb C}[X,Y,Z,W]/f(X,Y,Z,W)$ is a three-dimensional normal ring over ${\Bbb C}$. I am seeking for $f(X,Y,Z,W)$ such that ${\mathrm{Spec}}\,A$ has singularities along a curve $...
- 1,067
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Relationship between étale cohomology and group cohomology of the étale fundamental group
I am wondering to what extent and under which conditions there exists a generalization of the well-known relationship between the cohomology of the separable Galois group of a field and the étale ...
- 1,750
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Is it always true that $\mathcal{X}(\mathcal{O}_k) =X(k) \cap \prod_{v \in \Omega_k} \mathcal{X}(\mathcal{O}_v)$?
Let $k$ be a number field and $\mathcal{O}_k$ its ring of integers. Let $\mathcal{X}$ be a separated scheme of finite type over $\mathcal{O}_k$, and set $X = \mathcal{X} \times_{\mathcal{O}_k} k$.
Is ...
- 83
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Rational points and sections on a family of genus-3 hyperelliptic curves
I am interested in the proof or disproof of some conjectures about rational points and sections over $\mathbb{Q}$ for the following family of genus-3 hyperelliptic curves:
$$
C_t: f(x,a)\, g(x,a)\, h(...
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1
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Flat connections and Lie-algebra homomorphisms from Der to End
I'm reading a classical paper by Katz and Oda on the Gauss-Manin connections, and I have question on one of its claims.
Let $S$ be a smooth $k$-scheme and $\mathscr{E} \in \operatorname{QCoh}(S)$. Let ...
- 1,450
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Lifting algebra isomorphism via Nakayama
Let $R$ be a commutative ring with 1. Let $A,B$ be two $R$-algebras (associative with $1$ but not commutative), which are also free $R$-modules of rank $n$.
Assume there is an isomorphism $f:A\otimes\...
- 1,656
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1
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Must the spectral map $\operatorname{Spec}B\to\operatorname{Spec}A$ be surjective if the image contains all closed points of $\operatorname{Spec}A$?
Let $A\to B$ be an injective homomorphism of commutative rings. If the image of the spectral map $f:\operatorname{Spec}(B)\to\operatorname{Spec}(A)$ contains all closed points of $\operatorname{Spec}(...
- 2,783
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1
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53
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The closed immersions of formal schemes
Do closed immersions of formal schemes have the similar equivalent definitions as schemes?
To be specific, let $X, Y$ be two formal schemes, and $X$ is a "closed subscheme" of $Y$.
I think ...
- 133
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172
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Understanding the sign in the pullback of canonical under blow-up
I know this question has been asked in various forms on this site, but I haven't found the answer to my exact question.
For simplicity, let's say I'm blowing up a smooth surface $S$ at a point $p$. ...
- 573
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169
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When is $\operatorname{Spec}(B)\to\operatorname{Spec}(A)$ an epimorphism is the category of schemes?
This must has been asked several times in this cite, so I apologize in advance if this turns out to be trivial...
Let $A,B$ be commutative rings. We know that if $A\to B$ is an epimorphism in $\mathbf{...
- 2,783
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Why is the sequence $0\rightarrow\mathscr{O}_C(C)^\ast\rightarrow k(C)^\ast\rightarrow\rm Div(C) \rightarrow Pic(C)\rightarrow 0$ exact?
Definition
A curve over a field $k$ is a separated scheme $C$ of finite type over $k$ which is integral of dimension 1.
Let $C$ be a normal curve over a field $k$. A divisor is an element of the free ...
- 553
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1
answer
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The equivalence of category between $G$-sets and schemes étale over $K$
I am referring to Lemma 03QR in the Stacks project.
I believe that the left-hand-side category in the statement of Lemma 03QR should be "schemes finite étale over $K$". To show the second ...
- 649
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0
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49
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shift by [-2] in the Tate motive definition
I am studying Cycles, Transfers And Motivic Homology by Voevodsky chapter 5. In his lecture he defines the Tate motive by $M_{gm}{}^{\tilde{\, \,}}(\mathbb{P}^1)[-2]$ where $M_{gm}(X){}^{\tilde{\, \,}}...
- 63
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0
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136
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Miranda X.$2$.F.
Let $X$ be a compact Riemann surface. Let Bar: $\Omega^1(X)\to H^{(0,1)}_{\bar{\partial}}(X)$
by sending $\omega$ to the equivalence class of $\bar{\omega}$ is $\Bbb{C}-$linear, and $1-1$.
Attempt:
So ...
- 5,290
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0
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107
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Equivalence of finite generation of two section rings
I'm reading a proof about the finite generation of canonical rings in algebraic geometry and have a question about a specific step in the argument. The proof aims to show that for a normal projective ...
- 11
3
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2
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218
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Rational points on a hyperelliptic curve defined by a product of two quartics
I am interested in finding all rational points on the hyperelliptic curve
$$
C: f(x)\, g(x) = y^2,
$$
where
$$
f(x) = \left(625x^4 + 3100x^3 - 11344x^2 + 6200x + 2500\right), \quad
g(x) = \left(961x^4 ...
4
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1
answer
158
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Are stalks of a pushforward sheaf determined by stalks at the preimages?
Let $f : X \rightarrow Y$ be a continuous function between topological spaces. Let $F$ be a sheaf on $X$, and $y \in range(f)$ and $x$ be a preimage. Then, there is a natural map $(f_* F)_{y}\...
- 7,011
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1
answer
84
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If modular form satisfies $a_n(f_E) \equiv a_n(f_F) \pmod{p}$ for all $n \geq 1$, are the semisimplification isomorphic as $G_{\mathbb{Q}}$-modules?
Let $p$ be a prime. Let $E$ and $F$ be elliptic curves over $\mathbb{Q}$, and let $f_E$ and $f_F$ denote the associated modular forms. Suppose that all Fourier coefficients are congruent modulo $p$, i....
- 6,877
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2
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93
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Determine whether the image of $f(t) = (t^3,t^5)$ is an affine algebraic set
The problem is to prove whether the image of $f$ is affine over an algebraically closed field because the image of $f$ is contained on $V(x^5-y^3) =V $, so to prove the equality I tried to pick (x,y) ...
- 1,212