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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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Functions restricted to $\operatorname{Spec}(k[x,y]/(x^2,xy))$ with $k$ algebraically closed.

Consider functions $g\in k[x,y]$ restricted to $\operatorname{Spec}(k[x,y]/(x^2,xy))$ with $k$ algebraically closed.(i.e. $Spec(k[x,y]/(x^2,xy))\subset \operatorname{Spec}(k[x,y])$ allows such ...
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Projection Formula for Projective Morphisms

I have a question about a step in the proof of 5.3.17 in Liu's "Algebraic Geometry" (page 201): My question is why does Prop. 3.1.24 imply that $$\mathcal{O}_{SpecB} \cong q_*\mathcal{O}_{X_B}$$ ...
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Blow-up of affine space along subvariety

I was reading about blow-ups along sub varieties recently in Shafarevich's book and have a question concerning it. Let us take a curve $C$ in $\mathbb A^n$ and consider $X=Bl_C\mathbb A^n$. Since $C$ ...
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1answer
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Glueing T-structures

Suppose I have triangulated categories $A \xrightarrow{i_{*}} B \xrightarrow{j^{*}} C$ where $i_{*},j^{*}$ have right adjoints $i^{*}, j_{*}$ respectively (they possibly also have left adjoints). ...
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Factorization of morphism

This is relative construction in Mumford Algebraic Geometry 2 Chpt 1, Sec 7. Let $X$ be a scheme and $R$ is a sheaf of $O_X$ algebra. There is $Spec_X(R)$ scheme over $X$ s.t. for all $X-$scheme $Y$(...
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Why is it that $ V_{+}(I) \cup V(J) $ = $ V_{+}(I \cap J), $ where $ I $ and $ J $ are homogeneous ideals of a graded ring $ S$?

Define $ V_{+}(I) = \lbrace \mathfrak{p} \in \text{Proj}(S) \;|\; \mathfrak{p} \supset I \rbrace. $ I can see that $ V_{+}(I) \cup V_{+}(J) \subset V_{+}(I \cap J) $ since if $ \mathfrak{p} \in V_{+}...
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About algebraic variety

What is the difference between a variety of ideal and a variety of a set of polynomials, knowing that any finit set of polynomials generates an ideal,so why we define two kind? Are not these equal?. ...
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Question On Intersections of Homogeneous Prime Ideals

Let $ S $ be a graded ring, and let $ S_{+} $ denote the irrelevant ideal of $ S. $ And suppose that $ \mathfrak{a} $ is a homogeneous ideal of $ S $ contained in $ S_{+}. $ I am attempting to show ...
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1answer
40 views

Prove that the set of lines on smooth variety is a variety

Consider $X\subset\mathbb{P}^n$ smooth algebraic variety of degree $d$. I want to prove that the set of lines $F=\{l : l\subset X\}$ is a projective variety. As far as I understand, I need to show, ...
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Can someone give a closed immersion of schemes $f: Z\to X$ such that $f(Z)\subset U$ with $U$ a proper open subset of $X$?

Please give a closed immersion of schemes $f: Z\to X$ such that $f(Z)\subset U$ with $U$ a proper open subset of $X$.
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1answer
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$(S_f)_0$ is a finitely generated algebra if $S$ is. [duplicate]

Let $A, S$ be commutative rings with identity, and assume $S$ is a finitely generated $\mathbb{Z}^{\geq 0}$-graded $A$-algebra. If $f\in S$ is a homogeneous element of positive degree, $S_f$ is a $\...
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1answer
41 views

Criterion for Affine Scheme via Glueing

Let $X$ be a scheme. Futhermore we denote for a global section $s \in \Gamma(X,\mathcal{O}_X)$ the non vanishing set $$X_s := \{x \in X | s_x \neq 0 \}$$ (remark: $s_x$ is the image of $s$ under ...
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1answer
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Canonical bundle of blow up at singular point

Let $X$ be a complex variety/ manifold with one singular point $x_0\in X$. If we blow up $X$ at $x_0$, we obtain a smoot variety/manifold with exceptional divisor $Y$. How can we calculate the ...
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Completion of the local ring at a point on arithmetic surfaces.

Let $K$ be a number field and consider a arithmetic surface $X\to B=\operatorname{Spec} O_K$, i.e. $X$ is integral, regular, flat, proper over $O_K$ and it has dimension $2$. Now pick a closed point $...
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Understanding log complex for rational curve

I am trying to understand what exactly is $\Omega_X (log D)$ in a particularly case. More precisely, I am looking for conditions on a smooth surface and an effective divisor on it to obtain $\Omega_X (...
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Smooth proper variety over $\mathbb Q$ with everywhere bad reduction

$\newcommand{\Spec}{\operatorname{Spec}}$ It is a well-known fact that a smooth projective variety over $\mathbb Q$ has good reduction almost everywhere, i.e. everywhere apart from finitely many ...
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1answer
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How does the high school approach to classifying linear systems generalize to more variables, and why does it work?

If we have two linear equations in two variables, there may be $0$, $1$ or infinitely-many solutions. The standard approach to working out which case we're dealing with is probably to use Gaussian ...
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1answer
50 views

Lang Steinberg over separably closed field

Let $K=K^{sep}$ be a separably closed field with $K|\mathbb{F}_q$, where $\mathbb{F}_q$ is the field with $q$ elements. Let $\mathbb{G}$ be a connected linear algebraic group over $\mathbb{F}_q$. ...
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1answer
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$\tilde \phi:\Gamma (W)\rightarrow \Gamma(V)$ may not extends to all $k(W)$

A problem from Fulton's Algebraic Curves:-- Let $\phi:V\rightarrow W$ be a polynomial map between two affine varieties and $\tilde \phi:\Gamma (W)\rightarrow \Gamma(V)$ be the induced map between co-...
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How to prove that $\dim_k(k[x_1,\dotsc,x_n]/\sqrt{I})=|V(I)|$

How to prove that $\dim_k(k[x_1,\dotsc,x_n]/\sqrt{I})=|V(I)|$. I know that Hilbert Nullstellensatz will be required but I can't it out how?? With the notation common in algebraic geometry, the ...
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1answer
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Fulton Exercise 3.4 how to get double partial derivatives?

I have a question: $P =(0,0)$ is a double point on a curve $f(x,y) \in k[x,y]$ where $k$ is algebraically closed field. If degree of $f$ is $n$ and we write $f = f_n + f_{n-1} +......f_m $ where $f_{...
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Support for Fulton reading (chap 3 and 5) [on hold]

I am studying on my own the book of the Fulton (algebraic curves). My goal is to get to chapter 5. I happen to encounter difficulties in two chapters, 3 and 5. So I would like to know if there is ...
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19 views

Ample Sheaf and Affine Sets $X_s$

Let $X$ be a scheme and $L$ an ample sheaf. Futhermore let $s \in \Gamma(X,\mathcal{L})$ be a global section $s \neq 0$. My question is why and how to show that the non vanishing set $$X_s := \{x \...
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Why is $k[x_1] \to A$ not finite and $\phi(y)=x_1+x_2$ finite?

I was reading https://www.math.columbia.edu/~dejong/courses/deJongNotes.pdf. (Observe you don't have to look at the link as the question is self contained) See Example 1 in the beginning. There I ...
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21 views

Showing that two affine algebraic varieties are isomorphic

Let $W_1 = V(\langle u\rangle), W_2 = V(\langle v^2-u^3\rangle) \subset \mathbb{A}^2(\mathbb{C})$, $V_1 = V(\langle x, y\rangle) \subset \mathbb{A}^3(\mathbb{C})$ and $V_2 = V(\langle z, y^2-x^3\...
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Relation between tame symbol and residue on a curve

For an discrete valuation field $K$ we can define the tame symbol: $$(\,,\,)_K:K^\times\times K^\times\to \overline K^\times$$ $$(a,b)\mapsto(-1)^{v(a)v(b)}\overline{a^{v(b)}b^{-v(a)}}$$ Consider ...
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Showing the Kodaira Map is injective

I'm trying to prove the Kodaira embedding theorem via peaked sections, (exercise 7.10 in Székelyhidi's "An Introductionto Extremal Kahler Metrics"). My issue isn't to do with peaked sections, rather I'...
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1answer
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How to prove this preliminary result for Noether Normalization Lemma?

I need to prove that for an arbitrary field $K$, if $f \in K[x_1,\ldots,x_n]$ is a non-zero polynomial, then there exist $a_1,\ldots,a_{n-1} \in \mathbb N$ and $\lambda \in K$ such that if $$ f(y_1+...
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1answer
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Prove that any $d$ ($d - 1$)-spheres in $\Bbb{R}^d$ that intersect at three or more points have centers lying on a ($d - 2$)-plane

I want to understand why given $d$ ($d - 1$)-spheres in $\Bbb{R}^d$ that intersect at three or more points, then their centers must lie on a ($d - 2$)-plane. For $d = 2$ this doesn't tell you ...
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Spaces of matrices and projective spaces

Consider $\mathbb{C}^8$ i.e. 8 copies of the field of the complex numbers. I want to identify it with the set of matrices 2x4 $(z_{00},z_{01},z_{02},z_{03},z_{10},z_{11},z_{12},z_{13})$ now i have to ...
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Understanding of hyperplane defined by section.

In Hartshorne’s, a hyperplane $H_0$ in a projective space $\mathbb P^n$ can be defined by a global section $f\in \Gamma $($\mathbb P^n$,$\mathcal O_X(1)$), which is a linear polynomial. I have trouble ...
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1answer
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${(F^*)}^{-1}(m)$ is maximal ideal where $m$ is maximal

Our main objective is to interpret $F:V \to W$ a morphism as a map $F:maxSpec \Bbb C[V] \to maxSpec \Bbb C[W]$, $V,W$ are algebraic varieties. Now from $F:V \to W$ using Hilbert Nullstelensatz we ...
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1answer
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The existence of a linear map onto an affine algebraic set

Let $K$ be an algebraically closed field, $X\subset K^n$ be an affine algebraic set and $I$ be the ideal generated by all polynomials in $K[x_1,\cdots,x_n]$ that vanish on $X$. Prove that there ...
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Unnecessary premises in proposition about base change (Görtz-Wedhorn)

Here is proposition 4.20 from Görtz and Wedhorn's Algebraic Geometry I. It seems to me that the right square in (4.5.1) is completely unnecessary: We can always choose $S = X$ making the right square ...
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2answers
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“Cycles Premiers” In EGA

I am reading through EGA IV, 3.1, but I can't figure out what "cycles premiers" translates to. I don't know any french, really. I think it means "prime cycles", but an English search for that doesn't ...
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1answer
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About the tangent cone and tangent space of an affine variety

Let $X\subset \mathbb{A}^n$ be an affine variety then $X= Z(I)$ is the zero locus of the ideal $I$. In general the tangent cone at $0$ is define as $TC= Z(I^{in})$ where $I^{in}$ is the initial ideal ...
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1answer
66 views

Holomorphic form an a blow up

Consider the blow up of $\mathbb C^2/\mathbb Z_2$ at its singularity $0$. Since $dz_1\wedge dz_2$ is invariant under $z\mapsto -z$, it passes to a well defined holomorphic form on $(\mathbb C^2/\...
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1answer
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A question about morphism of projective spaces

Consider the morphism: $$ f: (\mathbb{P}^2 -\{(0:0:1),(0:1:0) \} )\to \mathbb{P}^3 $$ Given by $f((x:y:z))=(x^2:xy:xz:yz)$, my problem is to find the closure of the image of $f$, my argument was: ...
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Intersection of the strict transform of a curve and exceptional hypersurface in the blow up of the affine plane.

Consider the curve $Y$ to be $(\frac{t^3}{1-t},\frac{t^4}{1-t})$ with $t$ different from 1. I need to parametrize the curve, calling $x$ the first coordinate ad $y$ the second we find that $t^3=x(1-t)$...
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How to prove $\{G_i\to F\}$ is open covering only if $\forall$ field $K$, $F(Spec K)=\cup_iG_i(Spec(K))$?

This is an exercise in Eisenbud, Harris, Geometry of Schemes VI-11 as this part is skipped in Mumford Algebraic Geometry II. I think I figured out a way to do it but I am not totally sure. $\{G_i\to ...
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45 views

Is there a by-hand prove that $\Gamma(\mathbb{C}P^n,E)$ is finite dimensional for a holomorphic vector bundle $E$?

Let $E$ be a finite holomorphic vector bundle (or more generally a coherent analytic sheaf) on a compact complex manifold $X$. By the Cartan-Serre finiteness theorem, the cohomology $H^q(X,E)$ is a ...
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28 views

Affine Subsets of an (Abstract) Algebraic Variety are Open

Page 50. of this English translation of FAC says the following: "..if $V$ is an algebraic variety and $f$ is a regular function on $V$, we denote by $V_f$ the open subset of $V$ consisting of all ...
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Canonical section of a Hirzebruch surface

What is the definition of the canonical section of the Hirzebruch surface $\mathbb{F}_2=\mathbb{P}(\mathcal{O}(-2)\oplus \mathcal{O})$?
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Etale local systems and flat bundles

Over $\mathbb{C}$ (analytically) there is an equivalence of categories: Local systems $\iff$ vector bundles with a flat connection. Is this also true in the etale topology?
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Finite Dimensional $k$-Algebra has finitely many Maximal Ideals [duplicate]

I have a question about an argument used in Tamas Szamuely's "Galois Groups and Fundamental Groups" in following excerpt (see page 102): We start with affine variety $X$ of dimension $n$. According ...
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How to prove every holomorphic vector bundle on $\mathbb{C}P^n$ is an algebraic vector bundle

It is well-known that every holomorphic vector bundle on $\mathbb{C}P^n$ is an algebraic vector bundle, as a part of GAGA principle. Where can I find a reference for a detailed proof of the above ...
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1answer
59 views

Any Intersection of affine open subsets is affine

This is an exercise of Shafarevich's Basic Algebraic Geometry 1, Chapter 1 Section 5 Exercise 9 page 66 in the third edition: Show that any intersection of affine open subsets is affine [Hint: If $...
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1answer
48 views

When toric variety is affine?

Let $\Delta$ be a fan and let $X$ be a toric variety associated to $\Delta$. Is there a quick way to tell when $X$ is affine by looking at the fan? Thanks.
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1answer
105 views
+50

An open cover of $\mathbb{R}^n$ and $\mathbb{C}^n$

Consider the following subset of $\mathbb{R}^n$: \begin{eqnarray}V_i:=\{(p_1, \cdots, p_n)\in\mathbb{R}^n|x^n-p_1x^{n-1}+p_2x^{n-2}-\cdots+(-1)^np_n=0\text{ has at least one root with multiplicity at ...
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0answers
55 views

Resolution of singularity of $\mathbb C^2/{\mathbb Z_2}$ (blow up)

Consider the $\mathbb Z_2$-action $g:\mathbb C^2\to \mathbb C^2, z\mapsto-z$ on $\mathbb C^2$ and its quotient $X:=\mathbb C^2/{\mathbb Z_2}$. This is a singular surface with singular point the image ...