Mathematics

Questions tagged [projective-varieties]

351 questions
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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 projective variety $\tilde{X}$ and a surjective…
Zhen Lin
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Explicit example of a quintic threefold with 2875 distinct lines

A generic quintic threefold has 2875 lines. Is there an example of a quintic threefold explicitly defined by a polynomial that can be proven to have exactly 2875 distinct lines? I am particularly interested in an answer in the following form.…
Weiyan Chen
  • 111
9
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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 $\mathbb{P}^n$ by the homeomorphism $\varphi_{0}:…
Hetong Xu
  • 2,325
9
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4 answers

Isomorphism between projective varieties $\mathbf{P}^{1}$ and a conic in $\mathbf{P}^{2}$

I'm trying to establish an isomorphism between the projective line $\mathbf{P}^{1}$ and the conic in $\mathbf{P}^{2}$ defined by $Y=Z(g)$ , where $g=x^2+y^2-z^2$ . This is part of exercise I.3.1 in Hartshorne's Algebraic Geometry. I defined $\varphi…
JDZ
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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. Here is the outline: let X be our compact connected…
user880214
6
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1 answer

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 as quotients of polynomial functions on $C$ .…
red_trumpet
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What is the meaning of the residue field of a point in scheme?

If I consider the analogy of local ring at a point to the space of function germs at the point, then the residue field can be seen as the values that functions can take at the point. But when I consider the residue field of generic point or the…
Hypatia du Bois-Marie
  • 742
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Irreducible components of exceptional locus.

Let $\phi:X\rightarrow Y$ be a birational regular map between projective varieties where $Y$ is non-singular. Define $C=\{q\in Y:\dim(\phi^{-1}(q))>0)\}$ . Let $G=\phi^{-1}(C)$ . I saw the following statement: "Irreducible components of $G$ are…
Cusp
  • 218
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Problem in proving that $\mathbb{A}^2$ is not homeomorphic to $\mathbb{P}^2$

Let $k$ be an algebraically closed field. All spaces are equipped with the usual Zariski topologies. All the proofs of this fact that I've seen rely on the fact that two lines in $\mathbb{P}^2$ intersect but this doesn't necessarily hold in…
Luigi M
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Potential proof of Hartshorne Exercise I.4.7.

I've been trying to prove this exercise for a while, and I've finally got something that vaguely looks right. For reference, the question is: Let $X$ and $Y$ be two varieties. Suppose there are points $P \in X$ and $Q \in Y$ such that the local…
Fnark Man
  • 701
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Exact sequence obtained from a generic pencil on a projective surface

Let $X$ be a normal projective surface with only quotient singularities. I have a question about the proof of Theorem 10.8 (p.119) in Megyesi's Chern Classes of $\mathbb{Q}$ -sheaves . It goes as follows: Fix a projective embedding of $X$ . A…
blancket
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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 $U\subset U'\subseteq V$ with $U'$ of dimension…
Thomas Anton
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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 $X$ to be the closure of the union of all lines $P Q$ ,…
Eric
  • 463
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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 tensors $V^{\otimes n}_{\text{rank} = 1}$ is not…
QCD_IS_GOOD
  • 2,479
5
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1 answer

Analytification of a smooth projective variety is a compact Kähler manifold.

I am reading “Fourier-Mukai transforms in algebraic geometry” by Daniel Huybrecht. On page 130 it is written that by Hodge theory there is a natural direct sum decomposition $$H^n(X,\mathbb{C})=\bigoplus_{p+q=n}H^{p,q}$$ with…
Pouya Layeghi
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