Questions tagged [projective-varieties]
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About the definition of the morphisms of varieties in Hartshorne's algebraic geometry.
I am reading Hartshorne's algebraic geometry. According to the definition about morphism of varieties in Hartshorne's book, a morphism $\varphi:X \to Y$ should not only be continuous but also preserve ...
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lines passing through two points in $\mathbb{P}^n$
The reference is the Example 7.5. b) from ag notes by Gathmann. The paragraph explains the idea of projection from a point. He claims that the unique line passing through $a=(1:0:\cdots:0)\in\mathbb{P}...
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The definition for the morphism of varieties in Hartshorne's book.
I am reading the Hartshorne's algebraic geometry now, and by his definition for the morphism of varieties:
A continuous map $\varphi:X\to Y$, such that for every open set $V\subseteq Y$ and every ...
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Hyperplane section rationally equivalent to not lying on a hyperplane
Let $X \subseteq \mathbb{P}^n$ be a hypersurface and $L \subseteq \mathbb{P}^n$ be a generic plane of dimension $m-1 < n$. Then $X \cap L$ is a subvariety of dimension $n-m$ lying on the plane $L$.
...
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Hartshorne Chapter 1 Exercise 7.7 (a)
I am trying to solve part (a) of the following exercise of Hartshorne:
Let $Y$ be a variety of dimension $r$ and degree $d>1$ in $\mathbb{P}^{n}$. Let $P \in Y$ be a nonsingular point. Define $X$ ...
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The Graph of a Rational Map and Blowups
Let $X \subset \mathbb{P}^n, Y \subset \mathbb{P}^m$ be two projective irreducible varieties over $k=\mathbb{C}$ and
$$\varphi: X \dashrightarrow Y, \ \ Z:=[Z_0:Z_1:...: Z_n] \mapsto [A_0(Z):A_1(Z):.....
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Tangent developable of smooth complete intersection $V(f_1,..., f_k)$
Let $V=V(f_1,..., f_k) \subset \mathbb{P}^n$ be a complex variety (=separated scheme of finite type over $\mathbb{C}$) determined as vanishing set of homogeneous polynomials $f_i \in \mathbb{C}[x_0,.....
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Graph of Rational map
Let $X, Y$ be projective irreducible varieties over $k=\mathbb{C}$ and
$\varphi: X \dashrightarrow Y$ a rational map. Let $U \subset X$ the maximal open dense subset $U \subset X$ of points where $\...
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on projective toric varieties
I would like to understand better the embedding of projective toric varieties on projective spaces.
If I start with $X$ a projective topic variety, for being projective we know that there exists an ...
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Topology of the product of projective varieties
I can't prove that the closed sets of a product of projective varieties is a zero locus of multihomogeneous polynomials. I'm taking the abstract point of view to the construction of product of ...
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Bertini's Theorem in Harris' Algebraic Geometry
I have a question about the proof of Bertini' Theorem found in Harris' book Algebraic Geometry, on page 216/217:
Theorem 17.16. Bertini's Theorem. If $X$ is any quasi-projective variety over $\mathbb{...
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Disconnected fiber of regular birational map between complex projective varieties
Let $f: X \to Y$ be a regular birational morphism of projective varieties over $k= \mathbb{C}$, $q \in Y$ a point. Claim: If the fiber $f^{-1}(q)$ is disconnected, then $q$ is a singular point of $Y$ (...
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Functor of points of projective varieties (e.g., elliptic curve) over any commutative ring
Assume that $V \subset \mathbb P^n$ is a closed subvariety of a projective space over some field $k$. For instance $n=2, k = \mathbb F_5$ and $V : Y^2 Z = X^3 + Z^3$ is an elliptic curve.
For any $k$-...
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Degree of affine variety vs degree of projective variety
After reading this question on MO on the "degree" of an affine variety, I did not see why they said it was hard to give a lower bound on the number of intersection points of an affine ...
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Why algebraic variety is considered as scheme?
I'm a beginner of algebraic geometry, sorry to ask basic question.
I heard 'curve' over algebraically closed field $k$ is defined as 'integral separated scheme of finite type scheme over $k$' in terms ...
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Intersection of projective varieties
Is there any way to study varieties of the form below
$$
X=V\left(\{ f_i,g_j|i\in I,j \in J \right\}) \subset \mathbb{P}^{m+n+1},
$$
where
$$
f_i \in k\left[ x_0, \cdots , x_m \right],\ g_j\in k\left[...
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Intersection of schubert varieties
Let $L_1$ and $L_2$ $\in$ $\mathbb{P}^4$ be two planes that intersect in exactly one point $Q$. Let $P_1 \in L_1$, $P_2 \in L_2$ points, such that $P_1 \neq Q \neq P_2$. Using the duality theorem, ...
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How to present a projective variety in higher dimension space defined by the same equation?
I'm reading Hartshorne Ch1 and I'm wondering if there are some connection between projective varieties defined by the same equation in spaces with different dimension.
For example, we know that the ...
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Incidence correspondence of smooth hypersurfaces of degree $5$ in $\mathbb{P}^4$
I'm dealing with Exercise 11.11(a) in Gathmann's 2021 Notes of Algebraic Geometry, which is:
Exercise 11.11
(Let $K=\mathbb{C}$)
As in Exercise 10.23(b) let $U\subset\mathbb{P}^{\binom{4+5}{4}-1}=\...
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Intersection of (Toric varieties) and (Flag varieties)
For me, a variety $X$ is assumed to be irreducible and normal over some algebraically closed field $k$ of characteristic zero.
I call $X$ a toric variety if it has an algebraic torus $T$ which embeds $...
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Reduction of the Degree of a Curve by a Substitution
Let $y^2=P_{2n}(x)$ be an (hyper)elliptic curve, where $P_{2n}$ is a polynomial of degree $2n.$
It is said that the substitution $$x=x_1^{-1}+\alpha\qquad \text{and} \qquad y=y_1x_1^{-n}$$ reduces the ...
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Automorphism Group of rational normal curve of degree n
Let $V=\mathbb{C}^2$. $PGL_2(\mathbb{C})$ acts naturally on $V$. This action induces action of $PGL_2$ on $sym^n(V)$.
Let $\iota_n: \mathbb{P}(V)=\mathbb{P}^1\to \mathbb{P}^n=\mathbb{P}(sym^n(V))$ be ...
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Any finitely many points in $P^n$ can be included in an affine open chart
Let $\mathbb{P}^n$ be a projective space over an algebraic closed field $k$. I want to show that any finitely many points in $\mathbb{P}^n$ can be included in an affine open chart of it. A candidate ...
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Proof that Complex Affine Variety is a Psuedomanifold
When you first see the Nullstellensatz it's accompanied by a nice picture of an affine algebraic set over $\mathbb{R}$ in $2$ or $3$ dimensions. There's some flat bits, and some stringy bits, and they ...
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Projective closure of curve in $\mathbb{A}^2$
Let $k$ be an algebraically closed field. Consider the curve $V(y^2 = x^3 - 3x^2 + 2x) = C \subset \mathbb{A}^2$.
Now, we know that its the zero locus of some ideal, from what I gather this ideal is ...
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Show that every morphism $f\colon \mathbb{A}^1\setminus\{0\}\to \mathbb{P}^1$ can be extended to a morphism $\mathbb{A}^1\to \mathbb{P}^1$
Show that every morphism $f\colon \mathbb{A}^1\setminus\{0\}\to
\mathbb{P}^1$ can be extended to a morphism $\mathbb{A}^1\to \mathbb{P}^1$
This is Exercise 5.7(a) of Gathmann's 2021 notes of ...
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Find irreducible components of the following projective algebraic set
Consider the following projective algebraic set
$$\mathcal{V}(x_0^2+x_1x_2,x_0^2-x_1x_2+2x_3^2)\subset\mathbb{P}^3_{\mathbb C}.$$
I need to find the irreducible components. My approach has been the ...
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Non existence of homogeneous polynomials defining a morphism between projective varieties
Given the quadric $Q=V(XT-YZ)\subset\mathbb{P}^3$ and the lines $L_{X,Y}=V(X,Y)\subset Q$ and $L_{Z,T}=V(Z,T)\subset Q$, we have the morphism $\Phi: Q\rightarrow\mathbb{P}^{1}$ given by:
$\Phi(X,Y,Z,T)...
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How to find an isomorphism between two projecive subvarieties?
Consider the projective $2$-space $\mathbb{P}^2$ over the field $\mathbb Q$ given by the homogeneous cordinates $[X:Y:Z]$. Cinsider the following two elliptic curves in homogeneous coordinates: $$E_1: ...
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Normal bundle of fibers
Everything is defined over the field of complex numbers.
Let $Y \subset \mathbb P^{2n+1}$ be a smooth projective variety of dimension $n$.
Denote by $E$ the (naive) projectivization of the normal ...
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What is Chow's lemma really about?
By Chow's lemma, I mean any variant of the following basic result in algebraic geometry relating complete varieties to projective varieties:
Lemma.
For any complete variety $X$, there exist a ...
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How to show $x_0^2+x_1^2+x_2^2=0 \subset \mathbb{CP}^2 \iff \mathbb{CP}^1$
I am currently trying to blow-up an $A_n$ singularity defined by the hypersurface equation:
\begin{equation}
z_1^2+z_2^2+z_3^{n+1}=0 \subset \mathbb{C}^3
\end{equation}
Let $x_i, i=0,1,2$ denote the ...
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Why are meromorphic functions on a smooth projective curve rational?
Let $C \subset \mathbb P^n$ be a smooth connected projective curve over $\mathbb C$. Then the function field $k(C)$ consists of all functions $f$ which can locally (in the Zariski topology) be written ...
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multiplicity of the intersections between $p=x_0x_1^2+x_1x_2^2+x_2x_0^2\,\,$ and $\,\,q=-8(x_0^3+x_1^3+x_2^3)+24x_0x_1x_2$ in $\mathbb{P^2(K)}$
in $\mathbb{P^2(K)}$ where $\mathbb{K}$ is an algebraically closed field and $[x_0,x_1,x_2]$ the homogeneous coordinates, consider the following (homogeneous) polynomials:
$p=x_0x_1^2+x_1x_2^2+x_2x_0^...
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Real Projective plane $\mathbb{RP}^2$ [closed]
I visited this site about Real Projective plane $\mathbb{RP}^2$, or $\mathbb{P}^2\bigl(\mathbb{R}\bigr)$ if you prefer. My problem is this: when I implement the first equation of the Cross-capped disk ...
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What is a generically reduced scheme?
I am reading the book "3264 & All That
Intersection Theory in Algebraic Geometry". In the following definition (see page 30)
Definition 1.22. Let $f:Y\rightarrow X$ be a morphism of ...
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Describing all endomorphisms of elliptic curves
Let $k$ be an algebraically closed field. A curve is a separated integral $k$-scheme of finite type over $k$ and of dimension one. An elliptic curve $E$ is a smooth projective curve of genus one (a $k$...
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Degenerate plane conics form closed subset in $\mathbb{P}^5$ isomorphic to projection of $\mathbb{P}^2\times\mathbb{P}^2$
I am doing the following problem (Exercise 5.3.5) from "An Invitation to Algebraic Geometry" by Karen E. Smith et al.:
Show that the subset of all degenerate plane conics naturally form a ...
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Algebraicity of compact Riemann surfaces
I am taking a course in Riemann surfaces, in which the classical result about algebraicity of compact riemann Surfaces has been proven. However, I think there are some dubious points in the proof.
...
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Isomorphism between $Z(x_1^2-x_0x_2) \subseteq \mathbb{P}^2$ and $\mathbb{P}^1$
I want to show that if I have morphism $\varphi_0 : C_0 \rightarrow \mathbb{P}^1, \varphi_0([a_0:a_1:a_2])=[a_0:a_1]$ then I can extend it to be an isomorphism $\varphi : C:=Z(x_1^2-x_0x_2) \...
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Projection of rational normal curve is still a rational normal curve (of smaller degree)?
We work over $\mathbb{C}$. Let us define the rational normal curve of degree $d$ as the image of the morphism
$$\nu_d:\mathbb{P}^1\to \mathbb{P}^d,\quad [x:y]\mapsto [x^d:x^{d-1}y:\ldots:xy^{d-1}:y^d]....
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Fano Variety of a Hyperplane (Harris Algebraic Geometry A First Course)
I have a question about Example 6.19 (p. 70) in Harris' Algebraic Geometry: A First Course.
There is stated that the Fano variety
$$F_k(X) := \{\Lambda \in \mathbb{G}(k,n) \ \vert \ \Lambda \subset X \...
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Space of sections of adjoint varieties
We will work over the field of complex numbers and we will be consistent with the Grothendieck porjectivization, that is $\mathbb P(\cdot)=\operatorname{Proj}(\operatorname{Sym}(\cdot))$.
Let $X$ be ...
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Kempf's Proof that Projective Varieties are Complete
The following proof is taken from "Algebraic Varieties" by Kempf.
There seems to be way too many $n$'s in the proof. The projective variety is $\mathbb{P}^n$, the number of polynomials is $...
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Space of simple tensors
A simple tensor in $V^{\otimes n}$ is one that can be written as $v_1 \otimes \cdots \otimes v_n$ for some choice of $v_i \in V$, these are also called rank 1 tensors. The space of these simple ...
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How to prove this is an isomorphism of varieties
So I’m trying to prove all conics (i.e. zero sets of irreducible homogeneous polynomials of degree $2$) in $\mathbb{P}^2$ are isomorphic to $\mathbb{P}^1$ (here I work with classical algebraic ...
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Explicit computation for an elliptic curve.
Let $C\subset \mathbb P^2$ be the smooth curve $Y^2Z=X^3+Z^3$ and let $p:C_0\to \mathbb A^1$ be the projection $(x,y)\mapsto x$ from the affine part $C_0$ of $C$ (described by $y^2=x^3+1$) onto the $x$...
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Why is an elliptic curve with $j$-invariant in $K$ defined over $K$?
I often see in the literature some arguments like this:
"to show that an elliptic curve $E$ is defined over $\mathbb{F}_p$,
we show that the $j$-invariant of the curve is in $\mathbb{F}_p$."
...
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Support of homology in quasi-projective varieties.
Given a quasi-projective complex variety $X$ and a positive integer $i<\text{dim}(X)-1$. Consider the homology group $H_i(X(\mathbb{C}))$. Is it possible to find a subvariety of codimension at ...
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Two affine varieties which are isomorphic, but their projectivisations are not
I am learning Algebraic Geometry and came across the following question:
Show $V(y-x^3) \cong V(y-x^2)$ as affine varieties in $\mathbb{A}^2$. Prove that their projectivisations in $\mathbb{P}^2$ are ...
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