Questions tagged [projective-varieties]

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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\...
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Change of coordinates in projective space

I am doing exercise I.3.1. of Hartshorne, which asks us to prove that any conic in $\mathbb{P}^2$ is isomorphic to $\mathbb{P}^1$. After searching for some solutions, I found that almost everyone says ...
Sardines's user avatar
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Regularity of $H \cap X$ in Bertini's Theorem (Hartshorne CH. 2, Theorem 8.18)

In Hartshorne's proof of Bertini's theorem (CH. 2, Theorem 8.18), I am having difficulty in the first part of the proof which tells us that the scheme $H \cap X$ is regular at every point. I could ...
Sham astar's user avatar
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Finding a good homogeneous coordinate ring for a smooth projective variety

TL;DR Given a smooth projective variety $X\subseteq \mathbb{P}^m$ (I guess this means that the homogeneous coordinate ring $S(X)$ is regular whenever localizing at a non-maximal graded prime ideal). ...
Noto_Ootori's user avatar
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What are the conditions for a function $f\colon X \to Y$ to be regular when $X$ and $Y$ are prevarieties?

I'm currently following A. Gathmann's Algebraic Geometry, chapter 5 - varieties. I've seen the concept of prevarieties (ringed spaces with finite covers of affine varieties) and I'm stuck with the ...
Lucas Henrique's user avatar
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1 answer
94 views

Type of the intersection form of a degree $d$ hypersurface of $\mathbb{P}^3$.

Here is Exercise I.2 of R. Friedman's Algebraic Surfaces and Holomorphic Vector Bundles. Let $X$ be a smooth surface in $\mathbb{P}^3$ of degree $d$. Show using standard facts about the cohomology of ...
Display Name's user avatar
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Showing injectivity of a map between the sum of Picard groups of projective smooth irrudicible curve and the Picard group of the product of the curve

I want to solve the following exercise: Let $k$ be an algebraically closed field and consider a smooth projective irreducible curve $C$ over $k$. Set $S$ $:= C ×C$. Show that $p_{1}^* \oplus p^*_2 : \...
Lopi78's user avatar
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Is there a non-coherent subsheaf of $\mathscr{O}(-1)$ on $\mathbb{P}^1$

I'd like to ask that if there a non-coherent subsheaf of $\mathscr{O}(-1)$ on $\mathbb{P}^1$. I cannot think of an example.
Jean's user avatar
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How do I compute the action of an automorphism on the Néron-Severi group of a projective variety?

I am trying to read Dynamics of Automorphisms of Compact Complex Surfaces by Serge Cantat, and I am confused by his example of surfaces of degree (2, 2, 2) in section 2.4.6. The setup is the following:...
Zac's user avatar
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classification of morphism between varieties

Let $K$ be an algebraically closed field. We have different important theorems to study morphism of variety, in particular there is a clear and constructive description of the morphism between affine ...
Mario's user avatar
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3 answers
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If $f:X\rightarrow Y$ is a $k-$morphism of projective varieties and $X(k)\neq\varnothing$, then $Y(k)\neq\varnothing$

Suppose $X\subset \mathbb{P}^m,Y\subset\mathbb{P}^n$ are projective varieties defined over a field $k$, and that $f:X\rightarrow Y$ is a $k-$morphism, i.e. $f$ is a morphism which induces a $k-$...
Aaron Andersen's user avatar
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Finite morphism vs morphism of finite degree

Let $f:X \to Y$ be a morphism of projective varieties (integral schemes of finite type) over a field $k$. If $f$ is dominant, then $f$ induces a field extension $K(Y)\to K(X)$ of function fields. We ...
mathfan24's user avatar
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The canonical sheaf of hypersurface on $\mathbb{P}^n$

I want to prove the following result Let $X\subset\mathbb{P}^n$ be a smooth projective variety defined over $k.$ Let $Y= V(f)$ be a smooth subvariety of $X$ defined by a homogeneous polynomial $f$ of ...
ym2333's user avatar
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Defining algebraic varieties in general

I have encountered two general notions of algebraic variety when reading different texts in algebraic geometry, and wanted to ask whether they were equivalent or whether one is stronger than another. ...
user0134's user avatar
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$ (\mathbb{P}^n \times \mathbb{P}^n) \setminus \Gamma$ is isomorphic to an affine variety

We denote $\mathbb{P}^n$ to be the standard projective space over $\mathbb{C}$. Define $$\Gamma = \{ ([p_0 : ... : p_n] , [q_0 : ... : q_n]) \in \mathbb{P}^n \times \mathbb{P}^n | \sum_{i = 0}^n p_i ...
Steve's user avatar
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0 answers
63 views

There exist a hyperplane $H$ such that $\nu_d (\mathbb{P}^n) \cap H = \nu_d(\mathbb{V}(g))$

I am currently following an introductory course in algebraic geometry (I am 5 chapters in 'An invitation to algebraic geometry') and stumbled on the following problem (we denote $\mathbb{P}^n$ for the ...
Steve's user avatar
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4 votes
0 answers
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Intersection multiplicities of varieties expressed with curves

Let $X,Y$ be different irreducible projective varieties in $\mathbb P^n$ (over an algebraically closed field). Let $Z$ be an irreducible component of $X\cap Y$. Then the intersection multiplicity of $...
quantum's user avatar
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Is the ideal of projective variety homogeneous when the projective variety is defined over a ring

We call the set $\mathbf{I}(V) = \{f \in K [X] | f(x)=0,\text{ for all } x \in V\}$ the homogeneous ideal of projective variety $V$. Indeed, if $f$ and $g$ both vanish on $V$, and $r$ is an arbitrary ...
Zirui Yan's user avatar
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0 answers
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Bounding the degree of a variety after Segre embedding

Let $V \subseteq \mathbb{P}^{n_1}_k \times_k \ldots \times_k \mathbb{P}_k^{n_r}$ be a subvariety defined by multihomogeneous polynomials of multidegree at most $(d_1, \ldots, d_r)$, and let $v: \...
Andarrkor's user avatar
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0 answers
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Is $\mathbb{A}^{n+1}_\mathbb{C}\setminus \{0\} \to \mathbb{P}^n_\mathbb{C}$ a closed morphism?

I have already proved that the projection map $p \colon \mathbb{A}^{n+1}_\mathbb{C}\setminus \{0\} \to \mathbb{P}^n_\mathbb{C}$ given by $(x_0,x_1,..x_n) \rightarrow [x_0,x_1,..,x_n]$ is a morphism. ...
GendoTendoLendo's user avatar
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Geometry of the discriminant locus of a morphism between varieties

Let $f:X\to Y$ be a flat and proper morphism between two smooth algebraic complex varieties. The fibres $X_y$ are projective varieties of the same dimension $d$. The discriminant locus of $f$ is ...
manifold's user avatar
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Varieties with parameters

This seems like a very basic question to me and I am certain that people studied it a lot. It is for sure related to deformation theory and families of varieties, but I am not sure how these fields ...
Daniel W.'s user avatar
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0 answers
68 views

Defining an algebraic surface in terms of its parametrization

As part of a project in the history of mathematics, I am trying to read a book by Jung from the 1920s, Algebraische Flächen. Nevertheless, I am struggling to understand his set up already in the ...
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Example of a family of ample divisors $\{A_m\}_{m\geq 1}$ on a smooth projective variety $X$ such that $mA_m$ has a basepoint?

I know this famous example due to Kollár: Take $E$ an elliptic curve, and on $E\times E$ consider a horizontal fiber $F_1$, a vertical fiber $F_2$ and the diagonal $\Delta$. Let $X$ be a triple cover ...
imtrying46's user avatar
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1 answer
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Chow's Lemma proof in Milne's Book: Cartesian diagram

I'm reading the proof of the following statement in Milne's book "Algebraic geometry". Chow's Lemma: Let $V$ be a complete irreducible variety. There exists a projective variety $V'$ and a ...
Kandinskij's user avatar
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1 answer
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Chow's Lemma proof in Milne's Book: Locality of immersions.

I'm reading the proof of the following statement in Milne's book "Algebraic geometry". Chow's Lemma: Let $V$ be a complete irreducible variety. There exists a projective variety $V'$ and a ...
Kandinskij's user avatar
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1 vote
1 answer
87 views

Cokernel of $\mathcal{O}_m\to\mathcal{O}_n$ on $\mathbb{P}^1$

Consider the Serre twisted sheaf $\mathcal{O}_m$ and $\mathcal{O}_n$ with $m<n$ on $\mathbb{P}^1_k$ where $k$ is a field, as $\mathrm{Hom}(\mathcal{O}_m, \mathcal{O}_n)=\Gamma(\mathcal{O}_{n-m})$, ...
Jean's user avatar
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5 votes
0 answers
144 views

Is Every Closed Algebraic Set of Dimension $n$ Contained in a Closed Variety of Dimension $n+1$

Let $V$ be an algebraic variety of dimension $m$ over an algebraically-closed field of characteristic $0$, and let $n<m$ and $U\subset V$ be a closed subset of $V$. Must there exist a subvariety $U\...
Thomas Anton's user avatar
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2 votes
1 answer
88 views

Show the fibers are geometrically connected

This exercise is the 5.3.9 of Liu's famous book about algebraic geometry. Let Y be a normal, locally Noetherian, integral scheme, and let $f : X \mapsto Y$ be a projective dominant morphism with $X$ ...
Analyse300's user avatar
2 votes
0 answers
72 views

Potential typo in Fulton's "Algebraic Curves"

Could anyone verify whether the following is a typo? I'm studying William Fulton's "Algebraic Curves" and he's in the process of studying the relationship between affine space and ...
Ty Perkins's user avatar
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0 answers
58 views

Segre Embedding and lines on $x^2+y^2+z^2=1$ over $\mathbb{C}$

Let $x^2+y^2+z^2=1$ be the unit sphere over $\mathbb{C}$. Prove that the sphere has 2 rulings by straight lines. I have learnt the Segre embedding $\mathbb{P}^1(\mathbb{C})\times \mathbb{P}^1(\mathbb{...
Ishigami's user avatar
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0 votes
0 answers
50 views

Example of (maximal) projective spectrum

I read in a book the following definition of $Proj$. Given $A=\bigoplus\limits_{n\geq0}A_n$ a $\mathbb{N}$-graded $\mathbb{C}$-algebra without nilpotent, we can define $$Proj(A)= \text{homogeneous ...
wood's user avatar
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2 votes
0 answers
141 views

Composition of Gysin and restriction maps on $\ell$-adic cohomology

I follow the notations of Milne's lectures notes on etale cohomology, most specifically the section titled "The Gysin map" in chapter 24, p. 145. Let $k$ be an algebraically closed field, ...
Suzet's user avatar
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2 votes
1 answer
121 views

Does the completions $\hat{\mathscr{O}}_P(X)\simeq \hat{\mathscr{O}}_Q(Y)$ could deduce local rings $\mathscr{O}_P(X)\simeq \mathscr{O}_Q(Y)$?

In Hartshorne's algebraic geometry, he said the completion of local ring $\hat{\mathscr{O}}_P(X)$ takes much more 'local properties' than the local ring $\mathscr{O}_P(X)$. There are two natural ...
Frank's user avatar
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0 votes
1 answer
114 views

Compute the Zariski closure of a set

I have to compute the Zariski closure of the image of the following rational map: $f:{P}^2 \rightarrow \mathbb{P}^4$ $[x_0:x_1:x_2]\rightarrow [x_0x_1:x_0x_2:x_1^2:x_1x_2:x_2^2]$ I have already proved ...
Aron's user avatar
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0 answers
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Intersection of an arbitrary number of projective varieties

Let $X_1,...,X_r\subseteq \mathbb{P}^n$ be projective varieties of dimensions $i_1,...,i_r$. Are there any criteria to determine if $X_1\cap ... \cap X_r\neq \varnothing$? I know that if I have $X_1,...
Kandinskij's user avatar
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1 vote
1 answer
156 views

Is a morphism of varieties determined by its behaviour on an open subset

Let $\varphi:X\to Y$ be a morphism of varieties and let $X$ be irreducible. If $U\subseteq X$ is a non-empty open subset, is it true that $\varphi$ is the only extension to $X$ of $\varphi|_U$? I know ...
Kandinskij's user avatar
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1 vote
0 answers
50 views

Reducing elliptic curves -- why can we also reduce the endomorphisms?

Let $K$ be a local field with residue field $k$, let $E/K$ an elliptic curve of good reduction, $\tilde{E}/k$. The reduction map $E(K)\to \tilde{E}(k)$ respects addition, which leads to a natural ...
Mastrem's user avatar
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3 votes
1 answer
111 views

Trouble understanding Hartshorne's Algebraic Geometry Exercise 2.2 (Chapter 1).

I am trying to solve the following exercise: Let $\mathcal{a}$ be a homogeneous ideal such that $\mathcal a \subset S = K[x_0,\dots,x_n].$ Show that the following affirmations are equivalent: $Z(a) =...
xyz's user avatar
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0 votes
1 answer
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Generator of the maximal ideal of $\mathcal{O}_p$

I know that $x^3-xz^2-y^2z=0$ is a nonsingular curve in $\mathbb{P}^2$(char $k\neq 2)$. By definition, each local ring is a regular local ring. Consider the ring of degree zero elements in a ...
Display name's user avatar
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Is a nonconstant morphism of projective varieties necessarily finite?

Let $k$ be a field, and let $X$ and $Y$ be projective varieties over $k$. Do there exist morphisms $X \to Y$ that are neither finite nor constant? I know this cannot happen for $X$ and $Y$ curves (as ...
mathfan24's user avatar
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0 answers
205 views

Does birational equivalence between varieties preserves dimensions?

I assume that birational equivalence may not preserve dimensions, i.e. if there are two birationally equivalent varieties $X$ and $Y$ (birational maps $\varphi:X\to Y, \langle U,\varphi \rangle$ and $\...
Frank's user avatar
  • 141
1 vote
0 answers
117 views

Hausdorff dimension of an algebraic variety

I have the following elementary question that I cannot quite figure out myself or find an appropriate reference: Let $p$ be a nonzero homogeneous polynomial which we view as a function on $\mathbb R^n$...
Sobolev's user avatar
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1 answer
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Complete Intersection of Hypersurfaces are Fano varieties

I am currently studying Algebraic Geometry, and the wikipedia page of Fano Variety says the following "a smooth complete intersection of hypersurfaces in n-dimensional projective space is Fano if ...
user avatar
2 votes
1 answer
178 views

Intuition behind equations of Segre variety

I am a beginner at algebraic geometry. I am studying the Segre embedding $$\phi:\mathbb{P}^1\times \mathbb{P}^1\longrightarrow \mathbb{P}^3$$ sending $((x_0:x_1),(y_0,y_1))$ to $(x_0y_0:x_0y_1:x_1y_0:...
ABC's user avatar
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0 votes
1 answer
92 views

How to prove the image of the morphism $\varphi:C\to \mathbb{P}^1$ is dense in $\mathbb{P}^1$?

Consider an affine variety $C$ in $\mathbb{P}^2$ determined by $x^2+y^2-z^2=0$, then how to prove the image of the following morphism is dense in $\mathbb{P}^1$? $$\begin{align} \varphi:C&\to \...
Frank's user avatar
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1 vote
1 answer
156 views

Applications of the Lefschetz Hyperplane Theoren

There are a couple of applications of the Lefschetz Hyperplane Theorem I am struggling to wrap my head around. Hopefully someone knows how these facts are deduced directly from the theorem. Suppose $X$...
Shrugs's user avatar
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0 votes
1 answer
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Exercise 1.12 (c) of Silverman "The Arithmetic of Elliptic Curves"

I am self-studying "The Arithmetic of Elliptic Curves", but I am having difficulties with Exercise 1.12(c). The problem is as follows: Some definitions: The "only if" part is ...
Kevin's user avatar
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1 vote
1 answer
144 views

Gluing two affine curves to make a smooth curve

Let $C_0: y^{2}=x^4-7$ be a genus $1$ curve. Let another affine curve be $C_1: v^{2}=u^{4}(1/u^4-7)=1-7u^4$. The glueing maps between the two charts are given by $(x,y)\mapsto (1/x,y/x^{2})$ and  $(...
Pont's user avatar
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0 answers
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Smoothness of affine variety consisting of finitely many points

Let $R$ be a commutative ring such that $X:=\mathrm{Spec}(R)=\{p\}$, where $p$ is a prime ideal. Then, obviously, the affine variety $X$ consists of one point only. However, can we decide if this ...
Flavius Aetius's user avatar

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