Questions tagged [projective-varieties]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2
votes
0answers
78 views

How can I prove that affine hypersurface $V(X^2 + Y^5+Z^5 + 1) \subset \mathbb{A}^3$ is not rational?

$(*)$ I would like to prove that $Spec(\mathbb{C}[X,Y,Z]/\langle X^2+Y^5+Z^5+1 \rangle)$ is not rational (or equivalently that $Proj (\mathbb{C}[X,Y,Z,T]/\langle X^2 T^3+Y^5+Z^5+T^5 \rangle)$ is ...
1
vote
0answers
37 views

Image of Kähler class of an octic K3 surface is primitive and satisfies $p(\kappa) \cdot p(\kappa) = 8$

I am trying to understand the following sentence from a paper by Kovalev called Constructions of Compact $G_2$ Holonomy Manifolds. Unfortunately I can't find a copy of the paper publicly available so ...
0
votes
1answer
27 views

Free Abelian Group Generated by Codimension 1 Subvarieties: A Line.

i) A Weil divisor is a sum over the codimension 1 subvarieties, does that mean any point is generating the other points or are they all equally considered generators? ii) This paper says that the sum'...
0
votes
1answer
39 views

Using blowups to desingularise curves - understanding a simple example

I'm learning about blowups in Algebraic Geometry, and am having trouble understanding how to apply them to desingularise varieties. To illustrate my confusion, I will use the first example from these ...
3
votes
1answer
67 views

How should I understand Kodaira dimension?

Say we have projective variety $X$. Its Kodaira dimension $\kappa(X)$ is defined by the “growth exponential” of $P_d := \dim H^0(X,K_X^{\otimes d})$ with respect to $d$, i.e. $\kappa(X) := -\infty$ ...
0
votes
1answer
53 views

Why is the general case easier when proving that finite sets in general position may be described as the zero locus of quadratic polynomials?

I have a problem while studying the proof of the following theorem (Thm 1.4 in Algebraic Geometry, A first course by Joe Harris): If $\Gamma \subset \mathbb{P}^n$ is any collection of $d \leq 2n$ ...
1
vote
1answer
49 views

Degree of union is the sum of degrees

Suppose that $X$ is a reducible projective variety with equidimensional irreducible components $X_i$. Then I am trying to show that $\deg X = \sum_i \deg X_i$. This has been asked before. See here and ...
0
votes
0answers
43 views

Are the general and affine fundamental matrices nested models?

Given two models with the parameters in matrix form as $\mathbf{F}_\text{G} = \begin{pmatrix} f_{1} & f_{2} & a \\ f_{3} & f_{4} & b \\ c & d & e \\ \end{pmatrix} \hspace{...
2
votes
0answers
95 views

Some Examples of Sheaves on Varieties (Hartshorne's Exercise II.1.21)

I'm currently working on exercises in Hartshorne's Algebraic Geometry and coming up with some questions on Exercise II.1.21. I'll briefly restate the exercise and ask questions along with the ...
2
votes
0answers
66 views

Conic bundles structure

This question is all about paragraph $1$ of Sarkisov's article On Conic bundle structures. Here is a capture of the part I am interested in: Let $k$ be an algebraically closed field whose ...
1
vote
1answer
36 views

Bertini's Theorem Proof Explanation from Vakil's notes

I am trying to work through the proof of Bertini's Theorem from Vakil's notes. (sectin 12.4.3 to be precise). The setup is $X=V^+(f_1,f_2,\dots ,f_r)\subset \mathbb P^n_k$ is a smooth projective ...
1
vote
2answers
71 views

universal property of Albanese variety

Where can I find proof for the universal property of Albanese variety? (The universal property of the Albanese variety): For any (smooth projective) variety $X$ over a field $k$, there exists an ...
1
vote
1answer
38 views

Given a projective variety, finding another variety of appropriate dimension with empty intersection

Let $X$ be a projective variety of dimension $r$ in $\mathbb P^n_k$ , where $n>r+1$ and $k$ is an algebraically closed field. Then, can we always find a projective variety $Y$ (depending on $X$ of ...
1
vote
0answers
59 views

How to show the blowing down a ruled surface is projective

Let $X$ be an irreducible projective threefold, let $S$ be a ruled surface over a curve $C$, where $C$ has finitely many singular points. Let $\pi:X\rightarrow Y$ be a map such that $\pi(S)=C\subset Y$...
1
vote
1answer
67 views

projective space minus a closed point

Let $k$ be an algebraically closed field and let $\mathbb P^n_k=\text{Proj}(k[x_0,x_1,...,x_n])$ . If $n\ge 2$, and $p\in \mathbb P^n_k$ is a closed point, then can $\mathbb P^n_k\setminus \{p\}$ be a ...
2
votes
1answer
54 views

Relation between tautological line bundle and blow up at the origin

We can define the projective $n$-space $\mathbb{P}^n$ as the quotient of $\mathbb{C}^{n+1}\setminus \{0\}$ by the action of $\mathbb{C}^*$ with all weights equal to $1$. Moreover we can define the ...
1
vote
0answers
31 views

Bijective morphism that is biregular on every irreducible component

Consider a morphism $f:X\to Y$ between two (quasi-projective, say) algebraic varieties. Let $X_1,\dots,X_n$ be the irreducible components of $X$. Suppose that $f$ is a bijection, that $f(X_1),\dots,f(...
1
vote
2answers
120 views

The projective closure of the twisted cubic curve

I'm now reading Hartshorne's Algebraic Geometry and trying to solve Exercise 2.9(b). Let $Y$ be an affine variety in $\mathbb{A}^n$. Identifying $\mathbb{A}^{n}$ with the open subset $U_0$ of $\...
2
votes
1answer
40 views

Formula of intersection of homogeneous ideal

I found that I misread the concept about homogeneous ideal in the projective space in the Hartshorne's Algebraic Geometry (this book). For any subset $Y$ in the projective space $\mathbb{P}^n$,the ...
0
votes
0answers
15 views

show that $\mathcal I_{\mathbb A^n}(\phi(Y))^h=\mathcal I_{\mathbb P^n}(\overline Y)$

The map $\phi :D(x_i)\rightarrow\mathbb A^n$ is a homeomorphic. Suppose that $Y$ is a closed subset of the open subset $D(x_i)$ of $\mathbb P^n$ . The closure $\overline Y$ of $Y$ in $\mathbb P^n$ is ...
0
votes
1answer
26 views

Projective Closure of Morphism of Affine Varieties

Every affine variety $V$ has a unique projective closure $\overline{V}$, and there is an injective morphism $\iota : V \rightarrow \overline{V}$ given by something like $\iota(X_1,\ldots, X_n) = (X_1, ...
4
votes
1answer
128 views

How to understand group action especially Galois action on a scheme?

I read a lot of books and find none of them give explicit descriptions of group action on schemes. I am very confused now and have lots of questions. So I think these questions will be relatively long ...
1
vote
1answer
60 views

Degree of a finite morphism $f: X\to \mathbb P^1$ induced from a non-constant rational function in the Riemann-Roch space of a divisor

Let $X$ be a smooth projective curve over an Algebraically closed field $k$. Let $D=\sum_i n_i [P_i]\ge 0$ be an effective Weil divisor on $X$ where $P_i$ s are finitely many closed points of $X$. ...
2
votes
0answers
52 views

Relating $l(A),l(B)$ and $l(A+B)$ for Weil divisors on a smooth projective curve

Let $X$ be a smooth projective curve over an Algebraically closed field $k$. Let $k(X)$ denote its function field. If $A, B$ are Weil divisors on $X$ such that $A$ is effective (i.e. $A\ge 0$) , then ...
0
votes
0answers
75 views

Hilbert polynomial of the blow-up of a projection

Let $p:\mathbb{P}^r\dashrightarrow\mathbb{P}^{r-k-1}$ with $0\le k<r$ be the projection $(a_0:\cdots:a_r)\longmapsto (a_{k+1}:\cdots:a_r)$. Such a rational map has base $K:=V(X_{k+1},\cdots,X_r)\...
0
votes
0answers
21 views

Generating Degree of a subvariety

This is example 1.8.38 of Positivity in Algebraic Geometry I. The statement is that (after some translation) $I_X(d)$ is globally generated, where $X\subset \mathbb{P}^r$ is is a variety of degree $d$ ...
0
votes
0answers
60 views

Calculate homogeneous localization $k[x_0,x_1, x_2]_{((x_1-x_0, x_2-x_0))}$

[Definition] Let $S$ be a graded ring, and $\mathfrak{p}$ be a homogeneous prime ideal in $S$. Then we denote by $S_{(\mathfrak{p})}$ the subring of elements of degree $0$ in the localization of $S$ ...
0
votes
0answers
39 views

Problems regarding pullback of a Cartier divisor

We consider a Cartier divisor $D=div(f_i)$ on a irreducible projective variety $Y$ and let $f: X\to Y$ a morphism. I have the doubt it is not well defined in general a morphism $f^*: Div(Y)\to ...
1
vote
1answer
49 views

Is $\mathbb A^n(\mathbb Z)$ Zariski dense in $\mathbb P^n(\mathbb C)$?

Since $\mathbb A^1(\mathbb Z)$ is infinite, it is obvious that $\mathbb A^1(\mathbb Z)$ is dense in $\mathbb P^1(\mathbb C)$ with respect to the Zariski topology. Does the same property hold for $\...
1
vote
0answers
51 views

The twisted cubic curve in $\mathbb P^3$ [duplicate]

I am trying to solve exercise 2.9 b from section 1.2 of Hartshorne algebraic geometry. It asks to find out the generator of $I(\bar Y)$ where $\bar Y$ is the projective closure of the twisted cubic ...
0
votes
0answers
27 views

Ramifications indices invariant under algebraic closure + question about ramification and covering

We are considering functions fields of transcendance degree equal to $1$ (or smooth curves, it's the same), over a perfect field if necessary, or even a finite fields, let's say $K(X)$ and $K(Y)$ (...
2
votes
1answer
63 views

Problem in proving a statement regarding projective closure of an affine variety.

In problem $2.9$ of Hartshorne section $1.2$, he defined projective closure of an affine variety. Let $Y\subset \mathbb A^n$ be an affine variety, let $\phi : U_0 \rightarrow \mathbb A^n$ be the ...
1
vote
1answer
32 views

Clarification about Ideal and zero sets of empty set in Varieties

While defining affine and projective varieties we consider Zariski topology on $\mathbb A^n$ and $\mathbb P^n$. In the process we define $ Z(T)$, zero set of $T $ where $T\subset A=k[x_1,...,x_n]$ and ...
1
vote
1answer
68 views

Smooth $k$-varieties and geometrically irreducible smooth $k$-varieties over a finite field

I have a question about the equivalence between the smooth $k$-varieties (irreducible, by definition) and the algebraic functions fields with full field of constants $k$, up to birational equivalence, ...
0
votes
0answers
17 views

Equivalence between definitions of morphism of $k$-variety : Did I have understood well?

There is multiple definition of morphisms of projective $k$-varieties, and I want to make sure I'm okay with the equivalence between them. Actually, the definition I know for a morphism $\phi : V \...
1
vote
1answer
164 views

Branched cover in algebraic geometry

I've watched a lecture on K3 surfaces (although K3 surfaces are not the point of this question) where the following example is given: Let $\pi:S\stackrel{2:1}{\to}\Bbb{P}^2$ be the branched double ...
1
vote
1answer
71 views

Confusion about definitions of rational map between projective varieties

I've been learning some algebraic geometry from a combination of: Chapters 1 and 2 of Silverman's Arithmetic of Elliptic Curves, Reid's Undergraduate AG Hulek's Elementary AG and I'm a bit confused ...
1
vote
1answer
42 views

The function field of an affine part of a projective variety

Let $\phi:\mathbb{A}^n \to U_0\subseteq \mathbb{P}^n$ be given by $\phi(a_1,\ldots,a_n)=(1:a_1:\ldots:a_n).$ Let $X\subseteq \mathbb{P}^n$ be an irreducible Zariski-closed subspace (I call this a ...
2
votes
0answers
78 views

Hartshorne's Remark 7.8.2

This remark presents another form of proposition 7.3, in terms of linear systems. What troubles me is the second condition of this version. It's formulated as follows. (2) $\mathfrak{d}$ separates ...
0
votes
0answers
50 views

Looking For Sources Regarding Certain Riemann Surface Facts About Complex Tori

I just finished the NPTEL YouTube course on Riemann Surfaces, and I am looking for references where I might find complete proofs of the following four facts. I would like to emphasize the word ...
0
votes
0answers
85 views

Morphism from projective varieties

so I just started studying projective varieties (over algebraically closed fields) and I simply want to understand why $$V_{P_n}(T_i) \simeq P_{n-1}$$ whereas $V_{P_N}(T_i):= \{[z_0,..., z_n]\in P_n| ...
1
vote
0answers
17 views

On an analogy of the highest generating degree and reduction of ideals

Let $R=\mathbb C[x,y]$. Let $\mathfrak m=(x,y)$ . Let $J \subseteq \mathfrak m$ be a homogenous ideal with $\sqrt J=\mathfrak m$ i.e. $\mathfrak m^n \subseteq J$ for some integer $n\ge 1$. Let $a\ge ...
0
votes
0answers
56 views

Degree of morphism between affine varieties $= \#$ preimages in general fiber

Let $f: V \dashrightarrow W$ a rational dominant map between two affine algebraic $k$-varieties $V \subset \mathbb{A}^t, W \subset \mathbb{A}^s$ of same dimension. By category equevalence this is ...
1
vote
0answers
50 views

Correspondence Between Affine and Projective Varieties

Exercise 1.4.5.2 of Shafarevich's Basic Algebraic Geometry, Vol. 1, asks to prove that there is a one to one correspondence between affine closed sets in $\mathbb{A}^n _0$ and projective closed sets ...
0
votes
0answers
29 views

How to calculate the chow form of the projective closure of the product of affine varieties when given the chow forms of their projective closures.

Let $k$ be a field, not neccesarily algebraicly closed though I don't mind. Let $X\subset\mathbb{A}^{n},\;Y\subset\mathbb{A}^{m}$ be two algebraic sets (not neccesarily irreducible). I am interested ...
0
votes
0answers
37 views

Projective varieties: rational maps and morphisms of schemes

I'm a beginner with algebraic varieties topics, and I studied a (very little) of scheme before. The fact is that I don't truly manage to correctly understand the bridge between the different notions ...
0
votes
1answer
37 views

Hilbert-Samuel multiplicity of standard graded $k$-algebra which is an integral domain and $k$ is algebraically closed

Let $R=\oplus_{i\ge 0} R_i $ be a graded domain such that $R_0=k$ is an algebraically closed field, $R$ is finitely generated $k$-algebra and $R=k[R_1]$. Let $d=\dim R>0$. Let $\mathfrak m=\oplus_{...
0
votes
1answer
38 views

Understanding a line from Fulton's book regarding projective plane

Here's a paragraph from the book: How is he going from $z\neq 0$ to allowing $0$ in the last entry (e.g. $[1:m:0]$)? One way I can see is by taking the Zariski closure of the set since the set was a ...
0
votes
1answer
72 views

Rational points on projective varieties: what is density?

Dsiclaimer: this is a very basic question, but many books just skip this crucial point. Let $X$ be a smooth projective variety defined over a number field $K$. A closed point $x\in X$ is said ...
1
vote
1answer
52 views

On embedding $\mathbb C[x_1,…,x_d]/P $ inside $\mathbb C[[T]]$

Let $P$ be a prime ideal of $\mathbb C[x_1,...,x_d]$ such that ht$(P)=d-1$ i.e. $\dim (\mathbb C[x_1,...,x_d]/P)=1$. Then is it necessarily true that there exists an injective $\mathbb C$-algebra ...