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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Generalised tangent bundles of a complex variety.

Let $\Bbbk=\mathbb{C}$. Let $X$ be a $\Bbbk$-variety. Let $A$ be an Artinian local $\Bbbk$-algebra with residue field $\Bbbk$. I wonder whether the following is correct. There exists a $\Bbbk$-...
Display Name's user avatar
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Finding solution fby changing ellipses

Lets say we have two, $2$ dimensional positive integer vectors $X_0$ and $X_1$. We know that their sum is $[P,P]= X0+X1$. Lets also say assume that We get $Z=X1-X0$ from the intersection of $2$ ...
Aravind Muraleedharan's user avatar
1 vote
1 answer
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Do the bijections in the $(f^*, f_*)$ adjunction commute with restrictions to open sets?

$\newcommand{\ra}{\rightarrow}$ Let $f : X \ra Y$ be a morphism of ringed spaces. Let $F$ be a sheaf of modules on $X$ and $G$ be a sheaf of modules on $Y$. Then, there is a bijection $\alpha_{FG} : ...
David Lui's user avatar
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Expressing ideal sheaf of effective Cartier divisor as product of two ideal sheaves [duplicate]

If on a scheme, ideal sheaves $\mathscr{J}_1 \cdot \mathscr{J}_2 = \mathscr{I}$, where $\mathscr{I}$ corresponds to an effective Cartier divisor, is it necessary that $\mathscr{J}_1,\mathscr{J}_2$ ...
AprilGrimoire's user avatar
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1 answer
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For prime divisors $V,W\subseteq X$ in a smooth threefold $X$ and and integral curve $C\subseteq V\cap W$ with $i(V,W;C)>1$, do we have $V.C=W.C$?

Let $X$ be a smooth projective variety of dimension $3$ over an algebraically closed field. Let $V,W\subseteq X$ be prime divisors, i.e. two integral closed subvarieties of dimension $2$. Let $C\...
imtrying46's user avatar
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Points evenly populating n-dimensional space

What are ways to populate a bounded n-dimensional space with k points such that the points cover the space evenly? Evenly might be defined as: The areas of cells in the resulting Voronoi diagram have ...
Leon Derczynski's user avatar
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What are the cohomology groups $H^n(k, \mathbb{Q/Z})$ for a field $k$?

I'm reading a source that connects Brauer groups of fields $k$ to $H^1(k, \mathbb{Q/Z})$, but the source doesn't define the cohomology groups $H^\bullet(k, \mathbb{Q/Z})$. Googling the definition ...
mattematician 's user avatar
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Is it okay to explain Projective line with lines through origin in $\mathbb R^2$?

I want to start with definitions. Affine space $\mathbb A^n$ over the field $K$ is the set of $x_i$'s i.e., $\mathbb{A}^n = \{(x_1,\dots,x_n):x_i \in K \}$ if $n=2$, projective line defined to be $\...
Elise9's user avatar
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Map between Invertible Sheaves(not?) always injective

Let $X$ be a scheme and $\mathcal{M}, \mathcal{N}$ two invertible sheaves, i.e. locally free $\mathcal{O}_X$-modules of rank $1$, or equivalently that every $x \in X$ is contained in some appropriate (...
user267839's user avatar
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Rank of sheaf of differentials greater than dimension of fiber

I would like to prove the following statement (I'm not sure if this is even true): Let $f\colon X\to Y$ a morphism locally of finite type. For all $x \in X$ we have $$ \dim_{k(x)}(\Omega_{X/Y,x} \...
Patrick Perras's user avatar
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Why pullback of ideal sheaf should be the conormal sheaf?

I'm sorry that this isn't really a math question, but this gap between my intuition and the truth bothers me. For closed subvariety (for simplicity) with ideal sheaf $\mathcal{I}$, the pullback $i^*(\...
okabe rintarou's user avatar
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rational map by F. Mangolte

I'm reading Real Algebraic Varieties by F. Mangolte. Definition 1.3.22 (in the book) If $X$ and $Y$ are algebraic varieties over a base field $K$ a rational map $\phi:X\dashrightarrow Y$ is an ...
isz's user avatar
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Proving the $f^{-1}$, $f_*$ adjunction of sheaves directly

This question asks about proving the adjunction between $f^{-1}$ and $f_*$ for sheaves by showing that the unit and counit have the necessary properties by looking at stalks. I understand the answer ...
stillconfused's user avatar
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Quadratic equations defining a smooth hyperplane section of the 10-dimensional spinor variety. [closed]

Let $X\subset \mathbb P^{14}$ be a smooth hyperplane section of the $10$-dimensional spinor variety $S^{10}\subset \mathbb P^{15}$. Is it possible to describe $X$ as a scheme-theoretic intersection of ...
Kiwamu Watanabe's user avatar
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What is a... shtuka? [closed]

I´m trying to study Shtukas from Goss´ Basic structures of funcion field and from the Mumford paper. However, as expected, I don´t understand a lot of things. Is there a exposition talking about the ...
John F.'s user avatar
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Hartshorne Lemma III.2.8, exactness of direct limits of sheaves

In Lemma III.2.8 of Hartshorne, the author asserts that the direct limit of sheaves preserves exactness. I can't seem to find this earlier in the book, so I'm wondering if there is a general result ...
stillconfused's user avatar
4 votes
1 answer
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Zariski tangent spaces over residue fields

I came up with the following question while looking at the definition of Zariski tangent spaces. Let $R$ be a noetherian ring, and let $\mathfrak{m}\subset R$ be a maximal ideal. Write $k:=R/\mathfrak{...
Sardines's user avatar
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Checking isogeny between $E_1$ and $E_2$

This ss is from book The Arithmetic of Elliptic Curves, Joseph H. Silverman. Let $E$ and $E'$ be two elliptic curves defined over the field $K$. A non-zero morphism $\phi$ from $E$ to $E'$ such that $\...
Elise9's user avatar
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2 answers
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Why surjectivity implies isogeny is a finite map?

I want to start with some basics, Let $A$ and $B$ be two abelian varieties over $K$. An isogeny is a surjective homomorphism $\phi : A \to B$ with finite kernel. Isogeny is also defined for elliptic ...
Elise9's user avatar
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Differential 1-forms over an (hyper)elliptic curve

Given am elliptic curve $E: y^2=x^3+ax+b$, a nice basis for $\Omega^1$ (for the holomorphic 1-forms) might be $\{\frac{dx}{y}\}$. Given a hyperelliptic curve $H: y^2=P(x)$ where $P(x)$ is of degree $...
Or Shahar's user avatar
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Sheaf whose stalks vanish at closed points

I am self studying sheaves in algebraic geometry. I am wondering is the following is correct. Let $\mathcal{F}$ be a sheaf on a projective scheme $X$ such the stalk at any closed point $x$, $\mathcal{...
KAK's user avatar
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Algorithm to count adjacents of vertexes in a polyhedron

Is there an algorithm to count the adjacents of all the vertexes in a polyhedron defined by some linear equations? In other words, I have a $A$ by $N$ matrix $V_{ai}$ with $N<A$, and a dimension-$A$...
Wen Chern's user avatar
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Given any local homomorphism of local $k$-algebras, show that there is a morphism inducing the given local homomorphism.

Let $U$ and $V$ be quasi-affine varieties. Let $$\psi : \mathcal{O}_{V,q} \to \mathcal{O}_{U,p}$$ be the given local homomorphism of local $k$-algebra, where local homomorphism means that it is a map ...
glimpser's user avatar
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4 votes
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64 views

Two definitons of a pullback of a differential form

Let $f: X \to Y$ be a morphism of spaces with admitting differential forms (e.g. real manifold, complex manifold, smooth algebraic variety, schemes). Let $\Omega^n_Y$ denote the sheaf of $n$-forms on $...
CJ Dowd's user avatar
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Project of understanding proof that the ampleness of pull back implies ampleness under some condition ( Gortz's Algebraic Geometry book, Vol.2. )

I am reading the Gortz's Algebraic Geometry, Vol.2, proof of Lemma 23.7 and stuck at some argument Lemma 23.7. Let $X$ be a quasi-compact ( quasi-separated ) scheme, and let $i: X' \to X$ be a closed ...
Plantation's user avatar
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1 answer
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Deriative of algebraic implicit function is algebraic

Let $P(x,y)$ be a real polynomial with $P(0,0)=0$, and assume that $\partial_y P(x,y)$ is nonzero over $[-1,1]^2$. Then the equation $P(x,y)=0$ defines an implicit function $y=f(x)$ near $0$. In this ...
Tongou Yang's user avatar
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1 vote
3 answers
66 views

Point addition on elliptic curves

My question is about geometric interpretation of addition. It is known that if you have two points $A,B$ on elliptic curve $E$, draw straight line passing through $A$,$B$ and thanks to the Bezout ...
Fuat Ray's user avatar
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Locus of 4 colinear points in $\mathbb{P}^2$

I'm trying to solve exercises 2.34 and 2.35 in 3264 and all that (Eisenbud & Harris, 2016, p.80). Exercice 2.34 Let $\varphi\subset(\mathbb{P}^2)^4$ be the locus of 4-tuples of colinear points. ...
Ayoub's user avatar
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$\mathcal{O}_Y(-1)$ has no global section [duplicate]

Let $Y$ be a nonsingular closed subvariety of $\mathbb{P}_k^n$. It should be obvious that $\Gamma(Y, \mathcal{O}_Y(-1)) = 0$ but I can't prove it. I have two ideas: the first one is to calculate the ...
okabe rintarou's user avatar
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1 answer
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Geometric explaination of $I=\sqrt{I}$ implies $I$ has no embedded prime ideals

I'm working on the following exercise of Atiyah Commutative Algebra: Let $I$ be an ideal of a ring $A$ with $I=\sqrt{I}$ and has a primary decomposition. Show that $I$ has no embedded prime ideals. ...
MathLearner's user avatar
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Algebraic Geometry book for someone interested in Differential Geometry [duplicate]

I am very interested in differential geometry. However, in order to broaden my horizons, I would like to know the connection between algebraic geometry and differential geometry, any recommendations ...
Chengrui Han's user avatar
4 votes
1 answer
60 views

Reference request non existence of minimal resolution.

In this page of Wikipedia(https://en.wikipedia.org/wiki/Resolution_of_singularities), it writes, the hypersurface in $\mathbb{A}_\mathbb{C}^4$ defined by the equation $xy-zw$ has no minimal resolution....
George's user avatar
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Dlt pair is log canonical.

I am trying to understand why a dlt pair $(X,\Delta)$ is log canonical. So let $f\colon Y\to X$ a birational morphism and consider $E\subseteq Y$ a prime divisor. If $\operatorname{center}_X(E)\...
raisinsec's user avatar
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4 votes
0 answers
52 views

Deformation invariance of cohomology and Chern classes

Let $f\colon X\to Y$ be a smooth projective morphism between smooth complex quasi-projective varieties. Then it is known that each fiber of $f$ has isomorphic cohomology ring. I'm wondering if the ...
Kim's user avatar
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1 vote
1 answer
60 views

Degree 1 projective subvarieties are linear

I want to show that a closed subvariety (closed reduced equidimensional subscheme) $X \subseteq \mathbb{P}^n$ having degree $1$ is linear over an arbitrary field $k$, possibly non-algebraically closed....
Emory Sun's user avatar
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1 vote
1 answer
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Need Some Intuition on a Quotient Ring: $\mathbb{Z}[x,y]/(x^2+y^2-1)$

I've been diving into the world of algebraic structures and stumbled across something I'm trying to wrap my head around: the quotient ring $\mathbb{Z}[x,y]/(x^2+y^2-1)$. It essentially ties back to ...
Khaled Alekasir's user avatar
2 votes
1 answer
50 views

Remark in Atiyah, Macdonald: non-singularity $\Rightarrow$ analytic irreducibility

In Atiyah, Macdonald, Introduction to Commutative Algebra, Chapter 11, p. 124, after Proposition 11.24, there is a Remark. It follows from what we have said above that A is also an integral domain. ...
Elías Guisado Villalgordo's user avatar
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+50

Philosophy of applying Faltings' product theorem

Faltings' product theorem says that on $\mathbb{P}=\mathbb{P}^{n_1} \times \cdots \times \mathbb{P}^{n_m}$ over $k$ of characteristic $0$. For $\epsilon>0$, $d_1> \cdots >d_m$ decrease ...
finiteness's user avatar
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21 views

An Example of Flipping Contraction: the Contraction of Zero Sections of Vector Bundles?

Here is an example of flipping contraction: Consider the projective space $\mathbb{P}^d$ with vector bundle $\mathscr{E}:=\mathscr{O}(-a_1)\oplus\cdots\oplus\mathscr{O}(-a_r)$ for $a_i>0$. ...
WakeUp-X.Liu's user avatar
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Local coordinates of a simple normal crossing divisor

Let $A$ be a regular Noetherian local ring, and $\pi_1,\dots,\pi_r$ is a part of regular system of parameters, and so is $\varpi_1,\dots,\varpi_r$. My question is, if $\pi_1\cdots \pi_r = \varpi_1 \...
fyx1123581347's user avatar
2 votes
0 answers
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If $Y \subset X$ is a hypersurface so that $\mathcal{O}_X(Y)$ is positive, then $\operatorname{Pic}(X) \to \operatorname{Pic}(Y)$ is an isomorphism.

This is coming from exercise 5.2.10 in Huybrechts' book on complex geometry. The exact statement of the exercise is as follows. Let $Y$ be a hypersurface of a compact complex manifold $X$ with $\...
Daniel's user avatar
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Can a multivariate polynomial be rearranged into a univariate polynomial by applying only algebraic operations?

Let $\mathbb{K}\in\{\mathbb{Q},\mathbb{C}\}$ and let's consider polynomials over $\mathbb{K}$. If necessary, we can assume that the polynomials are $\mathbb{K}$-irreducible. My question is: Can a ...
IV_'s user avatar
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2 votes
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Which Leray acyclicity theorems are true, and when?

I've been doing some reading around the subject and I've encountered three (arguably five) variations of the Leray acyclicity theorem given by different sources. I know three of them are true; I'm ...
FShrike's user avatar
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2 votes
1 answer
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On $\dim \mathfrak{m}/ \mathfrak{m^2}$ for a local ring $(A,\mathfrak{m})$

I am trying to show that for every pair of integers $(s,t)$, there exists a Noetherian local ring $(A,\mathfrak{m})$ such that $s=\dim A \leq \dim(\mathfrak{m}/ \mathfrak{m^2})=t$ (the inequality is ...
l_btd's user avatar
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0 votes
1 answer
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geometrically understanding linear system $|2p|$ on an elliptic curve

Let $E$ be an elliptic curve (over the complex numbers), and let $p$ be a point on $E$. What does the linear system $|2p|$ look like geometrically? If $x \in \mathcal{O}(2p)$ is non-constant, then $...
usr0192's user avatar
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1 vote
1 answer
64 views

Cycle class of a smooth complete intersection $X \hookrightarrow \mathbb P^n$

Let $f:X \hookrightarrow \mathbb P^n$ be a $(d_1,\ldots ,d_r)$ smooth complete intersection over an algebraically closed field $k$. Let $\ell$ be a prime number different from the characteristic of $k$...
Suzet's user avatar
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4 votes
1 answer
80 views

What geometry is preserved by the translation maps on elliptic curves?

Let $E$ be an elliptic curve (over some field). For any $P \in E$, there is a translation map $T_P: E \to E$ given by $Q \mapsto P+Q$. This map is rational (i.e. the coordinates of $T_P(Q)$ are ...
popstack's user avatar
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0 answers
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+50

Construction of fibre product of schemes

So I understand the outline of the fibre product of schemes, but I am trying to fill in the details right now and am struggling. In particular, I am struggling with the step where $X$ and $Z$ are ...
Chris's user avatar
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How to prove the de Rham complex of sheaves of modules.

My question: Let $X$ be a topological space. Let $\mathcal{A} \rightarrow \mathcal{B}$ be a homomorphism of sheaves of rings. Denote $d: \mathcal{B} \rightarrow \Omega_{\mathcal{B} / \mathcal{A}}$ ...
jhzg's user avatar
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2 votes
0 answers
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Projective curve covered by two affine pieces

A non-singular projective curve $X$ is covered by two affine pieces (with respect to different embeddings) which are affine plane curves with equations $y^2 = f (x)$ and $v^2 = g(u)$ respectively, ...
ForgeBloyb's user avatar

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