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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Is an algebraic isomorphism between two smooth complex projective varieties also a symplectic isomorphism?

Let $X$ and $Y$ be two smooth complex projective varieties. So in particular they are Kähler manifolds, and hence we can consider them as algebraic varieties as well as symplectic manifolds. If $f:X \...
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15 views

“weak” Henselian property

I encountered following thread from MO treating the structure $X(k)$ of $k$-valued points of a separated algebraic space $X$ of finite type over $k$. I have a question about an aspect from Laurent ...
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Is pullback of ample sheaf by formal completion ample again?

Suppose $X$ is a Noetherian scheme and $I$ a quasi-coherent sheaf if ideals. We can formally complete the original scheme to get a formal scheme and a morphism of locally ringed spaces $k:\mathcal{X} \...
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12 views

Criterion for closed point in scheme of finite type.

Let $X$ be a scheme of finite type over an algebraically closed field $k$. Then there is a statement that: A point $x$ is closed if and only if the composition $k\to \mathcal{O}_{x,X}\to \mathbb{k}(x)...
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44 views

Is there a correspondence between $\operatorname{Spec}K[x_1,\dotsc,x_n]/\operatorname{Gal}(K/k)$ and $\operatorname{Spec}k[x_1,\dotsc,x_n]$?

The prime ideals of $\mathbb{C}[x]$ are $(0)$ and $(x-a)$ for $a\in\mathbb{C}$. Similarly, the prime ideals of $\mathbb{R}[x]$ are $(0)$, $(x-a)$ for $a\in\mathbb{R}$ and $(x^2+ax+b)$ for an ...
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Dimension of projection of projective variety on hyperplane

I have given a closed projective variety $X$ of dimension $k$ and a hyperplane $H$ in $\mathbb{P}^n$. When we take a point $P \notin H$ we can construct the projection $\pi$ by $P$ on $H$. I managed ...
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Quotients of Toric Varieties

This is a follow up of this question. Given a toric variety $X$ with a fan $\Sigma$ and a finite group $G$ acting on $X$, we know that the GIT quotient $X/G$ exists. However, as stated in the answer ...
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A Parametric representation of algebraic curves

Let $C$ be a plane curve in $\mathbb{C}^2$, defined by the polynomial $w^m+A_1(z)w^{m}+ \cdots + A_m(z)$, where $A(0)=0$ and $(w,z)$ is a local coordinate system of the $\mathbb{C}^2$ at the origin....
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The application of short-five lemma in the Hartshorne III 7.1

I am not sure with the last step when it applies the short-five lemma, which suppose to be the commutative diagram In order to apply the 5-lemma, we will need to show in addition (1) $\text{Hom}(\...
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24 views

Regarding the Proof of Hartshorne III.7.11

I got two questions regarding the proof: Why we can just assume $A_\mathfrak{m}$ as $A$ if we pick a sufficiently small neighborhood of $x$? (the 7-th line) Why's the last isomorphism true? Thank ...
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Motivation of separated scheme

A scheme is called separated if it is separated over $\mathbb{Z}$. Is there a specific reason to define like this, i.e. separated over $\mathbb{Z}$? I know Spec$\mathbb{Z}$ is a final object. But are ...
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27 views

Factoring map of curves over a field of characteristic $p$

I'm reading Silverman's book The Arithmetic of Elliptic Curves and I don't follow proof of a corollary. Corollary 2.12. Every map $\psi: C_1 \rightarrow C_2$ of (smooth) curves over a field of ...
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32 views

Group action on Artin ring

Let $k$ be a field and $A$ is a finite dimensional commutative $k$-algebra. Suppose a finite group $G$ acts on $A$ by automorphisms. Then $G$ acts on $\operatorname{Spec}(A)$. Suppose $A^G=\{a \in A | ...
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Why is a polarization of a torus called a polarization?

Consider a complex torus $T=\mathbb{C}^n/L$ where $L$ is a discrete subgroup of $\mathbb{C}^n$. We say the torus $T$ carries a polarization $H$ if $H$ is a positive definite Hermitian form on $\mathbb{...
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Methods for resolutions of sheaves on projective schemes

$\newcommand{\QCoh}{\mathsf{QCoh}} \newcommand{\mod}{\text{-} \mathsf{Mod}} \newcommand{\oh}{\mathcal{O}}$If I want to find a resolution of sheaves on an affine scheme $(X = \mathrm{Spec}(R), \oh_X)$ (...
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43 views

Self product $E \times E$ of an elliptic curve $E$.

Suppose that the self product $E \times E$ of an elliptic curve $E$ contains a compact Riemann surface $C$ of genus $2$. For points $a, b \in E$, we have to $D = a \times E+E \times b$ is ample ...
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Characterise polynomial roots as intersections of solution sets of its real and imaginary parts

Consider a polynomial equation $p(z)\equiv\sum_k \alpha_k z^k=0$ for $\alpha_k\in\mathbb C$. We can always understand this equation as a system of two polynomial equations, given by real and ...
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41 views

$A$ reduced implies $A\otimes_k K$ reduced, for $k$ perfect.

Let $k$ be a perfect field and $k\subset K$ any field extension. Let $A$ be any reduced $k$-algebra, if it helps we may assume it is finitely generated but the result should be true regardless. How ...
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20 views

$Nil(A)$ is prime implies $Nil(A\otimes_k K)$ is prime, for $k$ separably closed.

Assume $k$ is a separably closed field (i.e. the extension $k\subset\overline{k}$ is purely inseparable). Let $k\subset K$ be any field extension. Let $A$ be any $k$-algebra; if it helps we may assume ...
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1answer
35 views

On the hilbert polynomial of a coherent sheaf over a projective scheme

This is a lemma from the book by Huybrechts and Lehn. Let, $𝑋$ be a projective scheme over a field $\mathbb K$. Let, $\mathcal O(1)$ be an ample line bundle on $𝑋$, then the Hilbert polynomial $𝑃(�...
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21 views

Connection between intersection multiplicity definitions

Let $C$ and $D$ be complex plane affine algebraic curves defined by polynomials $f(x,y)$ and $g(x,y)$. One of the definitions for intersection multiplicity of $F$ and $G$ at some point $P$ is $$i(C,D,...
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Continuous map for closed points can't be extended to the whole projective space?

Given $X=\mathbb{P}^n_k$ for some algebraically closed field $k$ and some integer $n$. Let $Y$ be the set of closed points in $X$. Then it's said that a continuous map $f$ is defined on set $Y$ cannot ...
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30 views

Twisted cubic defined by one bihomogenous polynomial

The twisted cubic, $C$, is given by the image of the map $v_3: \mathbb{P}^1 \rightarrow \mathbb{P}^3$ defined as $$[x_0: x_1] \mapsto [x_0^3: x_0^2 x_1: x_0 x_1^2: x_1^3]$$ We can also see that $C$ ...
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18 views

Product of two closed immersions is also closed immersion

Let $f\colon X\longrightarrow Y$ be a morphism between $k$ - varieties. I Is the following statement true? If $f$ is a closed immersion, then $f\times f\colon X\times X\longrightarrow Y\times Y$ is ...
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1answer
21 views

Minimal ideal in the Affine Dimension Theorem

Let $Y,Z$ be affine varieties (irreducible). Then every irreducible components $W$ of $Y \cap Z$ correspends to the minimal prime ideals $\frak{p}$ of the principal ideal $(f)$. Why can we say that ...
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39 views

Blow-up, strict transform and tangent cone (Gathmann Notes, Exercise 9.22)

I'm studying Gatmann's Notes (version of 2014) https://www.mathematik.uni-kl.de/~gathmann/de/alggeom.php I'm currently reading the Chapter 9. Birational Maps and Blowing Up. I'm trying to do exercise ...
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23 views

How to give geometric structure to infinite union of affine schemes [duplicate]

It is a very known exercise in algebraic geometry that the union of countably many affine schemes cannot be an affine scheme. My question is, what is the most natural geometric structure we can give ...
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50 views

When is a plane algebraic curve closed?

When is a plane algebraic curve closed? Is there a metric? For quadratic http://mathworld.wolfram.com/QuadraticCurveDiscriminant.html we know the discriminant has to be below 0. Is there such a ...
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24 views

Codimension of a projective-variety $Y\subseteq \mathbb P^n_k$

Let $k$ be an algebraically closed field and $Y\subseteq \mathbb P^n_k$ be a projective variety. Let $I(Y)$ be its homogeneous defining ideal in $k[x_0,...,x_n]$ . Then is it true that codim$(Y)=ht (...
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44 views

Looking for some computable examples of sheaf of total quotient ring

This is mentioned in Ueno's Algebraic Geometry 3, Chpt 7, Sec 2 without concrete example. It is located at pg 43 right above exactsequence (7.33). Let $X$ be a scheme and $U=\operatorname{Spec} R\...
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what is the meaning of stratification? I have seen the definition of a topologically stratified space.

what is the definition of stratification in moduli space of stable maps? Is it the same as a topologically stratified space? For example, a stratification of moduli space of stable maps.
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Uniqueness of line in projective plane only intersecting conic in one point

Let $F$ be any field and let $E$ be its algebraic closure. Assume that $p\in F[x,y,z]$ is homogeneous and of degree two, and further, assume that $p$ is irreducible in $E[x,y,z]$. Then, how do I show ...
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How to understand this sheaf $\mathscr{O}_C(d-3)(-\Delta)$

Let $C\to C'$ be the normalization of some nodal plane curve $C'\subseteq\mathbb{P}^2$ and $\Delta$ be the divisor of preimage of nodes. Then $\mathscr{O}_C(d-3)$ is the twisted structure sheaf. Now, ...
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1answer
28 views

Relation of regular map with the map of tangent Cones

I am reading Chapter 1 of J.S. Milne notes. Link is here: https://www.jmilne.org/math/CourseNotes/LEC.pdf I am confused on Example 2.7(a) on page 18 of the notes. So the situation is we have a ...
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1answer
25 views

Linear Equivalence of Divisors Implies Algebraic Equivalence

The highlighted sections are taken from Volume 1 of "Basic Algebraic Geometry" by Igor Shafarevich. Definition: Let $ X $ and $ T $ be two arbitrary irreducible varieties. For any point $ t \in T ...
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Adjunction formula and Poincare residue

$\phi:C\to \Gamma$ be the normalization of a nodal plane curve. $\Delta$ is the divisor of the preimage of nodes. $\phi^*D$ to be the zero divisor of the mero- morphic function $\phi^*(g)$ on $C$. ...
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Exterior power of a locally free sheaf twisted

Let $X$ be a smooth projective scheme and $F$ a locally free sheaf of $\text{rank}(F) = r$ and $L$ a line bundle on $X$. This week here during a seminar, there was heated discussion about a result. ...
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15 views

A lattice polytope P is normal if and only if the polytopal semigroup S is normal.

We have $P\subset Z^n$ a lattice polytope and an additive semigroup $S=\{(u,m)\mid u\in mP\cap Z^n, m\in Z\}$, or equivalently $S$ is generated by $(\{1\}\cap P)\cap Z^n$. The definitions are: P is ...
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24 views

Flat pullback of divisors and invertible sheaves

Assume that $f:X\to Y$ is a flat, proper map of Noetherian schemes. In this setting I can define the pullback $f^\ast$ of invertible sheaves on $Y$ and of divisors (intended as cycles) on $Y$. Now ...
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On Gieseker stability of a coherent sheaf

Let, $X$ be a projective scheme over $\mathbb C$ with fixed ample dividor $H$ and $E$ be a pure coherent sheaf of dimension $=dim(X)$ on $X$ . Let, $q \in \mathbb Q[t]$ be a fixed polynomial with ...
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1answer
24 views

Independent conditions of linear system

I am reading the book on "geometry of algebraic curves" by Harris ect. And I met this statement here: My question is what it means that the conditions are independent, i.e., what I need to show here? ...
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1answer
17 views

Pushforward of an invertible sheaf under certain hypothesis on the Higher direct images

Assume that $f:X\to Y$ is a flat and proper morphism between intergral noetherian schemes. Assume that $L$ is an invertible sheaf such that $R^i f_\ast L=0$ for $i>0$. Can we conclude that $f_\...
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31 views

how to show that a map is an injection in a diagram of k - vector spaces?

Can anyone offer any help? I don't ask for the solution but can you give me a hint to start Let $X$ be a variety and $d$ a positive integer. Assume given for all $i$ $\in$ $I$ := $\{$1,...,$d$$\}$ ...
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59 views

Projection onto hyperplane is a closed subvariety?

This is a problem in algebraic geometry that I'm trying to figure out: Problem: Let $X$ be a subvariety of $\mathbb{P}^n$ of dimension $k$, with $1 \leq k \leq n-1$. Let $P$ be a point in $\mathbb{P}^...
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1answer
49 views

Showing set of of monic polynomials with repeated roots is an algebraic subset of $\mathbb{A}^n$.

Problem: Prove that the set $M$ of monic polynomials with repeated roots is an algebraic subset of $\mathbb{A}^n$. (throughout we work over an algebraically closed field $k$) Attempt: I'm a bit ...
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32 views

quadratic twists of $P_{\mathbb{Q}}^1$

I have seen somewhere else that if $C$ is a conic in $P_{\mathbb{Q}}^2$,then it is a quadratic twist of $P_{\mathbb{Q}}^1$. So I want to know how to define the quadratic twist of $P_{\mathbb{Q}}^2$(...
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53 views

Describe the associated ideals of these algebraic subsets of $SL_n$

Let $n$ be a positive integer, and let $M_n$ be the set of all $( n \times n)$ matrices with coefficients in an algebraically closed field $K$. Identify $M_n$ with the affine space $\mathbb{A}^{n^2}$. ...
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59 views

Def of Sheaf on Scheme

In hartshorne, for a ring $A$, the sheaf $\mathcal{O}$ on Spec $A$ is defined to be a map sending each open set $U$ of Spec $A$ into the set of functions $s: U \to \coprod _{p \in U} A_\frak{p}$ such ...
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2answers
44 views

Is $\Bbb P^n_k$ irreducible?

Is $\Bbb P^n_k$ irreducible? I could show that $\Bbb A^n_k= Spec k[x_1,\ldots, x_n] $ are irreducible, for all $n \ge 0$, field $k$. But it deosn't seem clear to me that irreducibility remains under ...
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57 views

Question about linear algebra in Benson's book “Representations theory of elementary abelian groups and vector bundles”

The following proposition is in Benson's book: Proposition 4.1.2 Suppose $k$ is algebraically closed. Let $A_1,A_2\in M_{n\times m}(k)$, regarded as maps from $k^m$ to $k^n$. Then: (1) Suppose ...