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Questions tagged [algebraic-geometry]

The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. This tag should not be used for elementary problems which involve both algebra and geometry.

The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. This tag should not be used for elementary problems which involve both algebra and geometry.

29074 questions
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Divisor of a global section of a line bundle associated to a Weil divisor

This is a simple question. Let $D$ be some Weil divisor on a non-singular projective variety $V$, $\mathcal{O}(D)$ the associated line bundle. Suppose $s\in H^0(V,\mathcal{O}(D))$ is a global section. How $div(s)$ and $D$ are related? For example on…
Benji01
  • 91
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Find Zariski closure of a set

Let $X=\{(x,\sin(x)): x \in \mathbb{A}^{2}\}$. I want to find the closure (with respect Zariski topology) of $X \subseteq \mathbb{A}^{2}$. OK I've already shown that $X$ is not a closed set. Now consider $cl(X)$ this is a closed subset of…
user10
  • 5,688
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Smooth cubic surface in $\mathbb{CP}^4$ is covered by lines

How to prove that smooth cubic surface $X$ in $\mathbb{CP}^4$ is covered by lines and the normal bundle of the generic line $l$ is $N_{l/X}=\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}$? $\textbf{Edit}$ Let me give some…
guest31
  • 925
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4 answers

Surjective+finite-type+quasi-finite doesn't imply finite

Exercise II.3.5 (c) in Hartshorne, Algebraic Geometry, asks to find an example of a surjective, finite-type and quasi-finite morphism of schemes which is not finite. I need to find a finitely generated $A$-algebra $B$ which is not finite generated…
A. Prufrock
  • 407
8
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1 answer

maps with connected fibers

Let $\pi: X \to S$ be a morphism of schemes. I will say $\pi$ is "pseudoconnected" if $\mathcal{O}_S \to \pi_* \mathcal{O}_X$ is an isomorphism (this is not standard language). If $\pi$ is proper with connected fibers, can we deduce that $\pi$ is…
user29743
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1 answer

Diagonal in projective space

This is exercise $2.15$ from Harris book "Algebraic Geometry: A First Course". Show that the image of the diagonal in $\mathbb{P}^{n} \times \mathbb{P}^{n}$ under the Segre map is isomorphic to the Veronese variety $v_{2}(\mathbb{P}^{n})$. Would the…
user10
  • 5,688
8
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2 answers

If $Y$ is a quasi-affine variety, then dim$Y$ = dim$\overline{Y}$

Reading through the proof of proposition 1.10 in Hartshorne's Algebraic Geometry I found some of it to be unnecessary. Is the following proof correct or can you point out my flawed logic? Let $Z_0 \subset ... \subset Z_n$ be a sequence of distinct…
Michael N
  • 355
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Motive of a curve and its Jacobian

Let $C$ be a smooth projective curve with a $k-$rational point $x_0$ and $J$ its Jacobian variety. Let us consider the (almost) canonical embedding $j:C \to J$ that sends $x_0$ to the identity $e \in J(k)$. There is a decomposition of the Chow…
Stefano Nicotra
  • 371
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2 answers

Is there error in proof of lemma on Riemann-Roch space of divisor $D$?

I'm reading Steven Galbraith "Mathematics of Public Key Cryptography" and can't understand lemma 8.4.2 on page 154 that necessary for proof of Riemann-Roch theorem. Suppose $C$ $-$ curve over $\mathbb k$. We want to prove that if $D' \ge D$ then…
petRUShka
  • 182
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votes
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Fibers of a scheme over $\text{Spec}\,\mathbb{Z}$

How to construct connected scheme $X$ over $\text{Spec}\,\mathbb{Z}$ such that for $p\neq0$ the fiber $X_p=X\times_{\text{Spec}\,\mathbb{Z}}k(p)$ over the prime ideal $(p)$ contains precisely $p$ points over $\mathbb{F}_p$? The same question for…
miguels
  • 165
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1 answer

Can a birational morphism surject from an affine to a projective variety?

Let $X$ be an affine variety and $Y$ a projective variety, both integral (reduced and irreducible). Assume that $\phi:X\to Y$ is a birational morphism. I would venture to say that $\phi$ can not be surjective, but I have no proof for the statement.…
Jesko Hüttenhain
  • 14,792
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On a certain morphism of schemes from affine space to projective space.

I'm currently reading Ravi vakil's notes on algebraic geometry, and I'm completly stuck on a question and I would love for you all to shed some light on it. The question is: "Make sense of the following sentence: '' $$\mathbb{A}^{n+1}_k \setminus…
Dedalus
  • 3,940
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votes
1 answer

Trying to parse a definition in Silverman's EC book

Let $C_1\subset \mathbb{P}^{N_1}$ and $C_2\subset \mathbb{P}^{N_2}$ be two curves. Then a map $\phi:C_1\to C_2$ can be defined as $$\phi=[f_0,\ldots,f_{N_2}],$$ where each $f_i$ is a homogeneous polynomial. Such a map induces a map $\phi^*:K(C_2)\to…
pki
  • 3,743
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0 answers

Show that the set of unitary matrices is not an affine algebraic variety in complex space $C^{n^2}$.

This is an exercise from An Invitation to Algebraic Geometry by Karen Smith. It asks to show that the set of unitary matrices $U_n$ is not an affine algebraic variety in complex space $C^{n^2}$. However it is a real algebraic variety. The second…
KittyL
  • 16,965
8
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2 answers

Are birational morphisms stable under base change via a dominant morphism

Let $f: X \to Y$ be a birational morphism of integral schemes and $g: Z \to Y$ a morphism of integral schemes which maps the generic point of $Z$ to the generic point of $Y$, i.e., the morphism $g$ is dominant. Is then $X \times_Y Z \to Z$…
user5262
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