# Questions tagged [projective-varieties]

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### 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 ...
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### 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 ...
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### 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'...
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### 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 ...
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### 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. $\kappa(X) := -\infty$ ...
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### 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 $d \leq 2n$ ...
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### 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 ...
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### 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 ...
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### 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 $X$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...