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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.

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Metric spaces as schemes

In Récoltes et Semailles, Grothendieck explains he invented the concept of scheme to unify algebraic varieties on algebraically closed fields, arithmetical varieties in characteristic $p$, and also the usual metric spaces. This last part is not…
V. Semeria
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Definition of formal neighbourhood

Consider the scheme $\mathbb{P}^1$, and the point $0 \in \mathbb{P}^1$. What is the formal neighbourhood of $0$ in $\mathbb{P}^1$? Or if you know a good reference, that would be helpful.
Puraṭci Vinnani
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Question regarding Hartshorne Example II.(6.5.2)

Let $k$ be a field, let $A=k[x,y,z]/\langle xy-z^2\rangle$ and let $X=\operatorname{Spec}A$. Let $Y:y=z=0$ I want to know the divisor of $y$ In Hartshorne book, because $y=0 \Rightarrow z^2=0$ and $z$ generates the maximal ideal of the local ring at…
Sang Cheol Lee
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Geometric meaning of the product of ideals

Consider the partial order of quasi-coherent ideals of a scheme $X$. Actually it also carries a product which is compatible with the partial order. (This makes this partial order a commutative quantale. See MO/26607 for the significance of the…
Martin Brandenburg
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Hartshorne exercise 1.6.4 : Is it true that $\mathcal{O}_{P,X} \cong \mathcal{O}_{\varphi(P),\Bbb{P}^1}$?

Let us work over a fixed algebraically closed field $k$ and consider a non-singular projective curve $X$ and $\varphi : X \to \Bbb{P}^1$ a non-constant morphism. My question is: For $P \in X$, do we have an isomorphism $$\mathcal{O}_{P,X}…
user38268
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Why 7 points on a twisted cubic is enough to fix a quadratic?

From Joe Harris, Algebraic Geometry, Page 10. Show that if seven points $p_{1},\cdots,p_{7}$ on a twisted cubic curve, then the common zero locus of the quadratic polynomials vanishing at the $p_{i}$ is the twisted cubic. I am looking for a hint…
Bombyx mori
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Why are singular conics reducible?

I'm currently working through Rick Miranda's book on Algebraic Geometry and Riemann Surfaces, and I've been stuck on a problem in the first chapter, and I can't seem to get anywhere. I think that for example Bezout's theorem would solve it, but I…
Dedalus
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Smooth Quartics in $\Bbb{P}^3$

Algebraic category. Ground field $\Bbb{C}$. This is a naive question: are all smooth quartic surfaces in $\Bbb{P}^3$ isomorphic ? The answer is NO if and only if there is a smooth quartic in $\Bbb{P}^3$ containing some (-1)-curve.
Heitor Fontana
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Why is the multiplicative group of a field an algebraic group?

Reading from here: https://en.wikipedia.org/wiki/Algebraic_torus In particular, the paragraph under the heading "Multiplicative group of a field." So, in my mind, a multiplicative group of a field is denoted $F^\times$ and is just the group $(F…
user637978
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The Unit Circle over $\mathbb R$ is not a UFD

This is exercise 14.2 N in Vakil, self-study. Similar questions have been asked a few times on this site, but none of the answers use a method that I believe Vakil intended: here, here, and here. We are to show $\mathbb R[x, y]/(x^2 + y^2 -1)$ is…
Johnny Apple
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Exercise 2.4 Fulton's Algebraic Curves

I am looking at exercise 2.4 in William Fulton's "Algebraic Curves". It asks to prove that if $X\subset \mathbb{A}^n$ is nonempty affine variety, then the following are equivalent $X$ is a point $\Gamma(X)=k$ $dim_k\Gamma(X)<\infty$ I have a…
Moss
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Hartshorne Ex. 1.3.8 - Where do I take intersections here?

Let $H_i$ and $H_j$ be the hyperplanes in $\Bbb{P}^n$ defined by $x_i = 0$ and $x_j = 0$ with $i \neq j$. I want to show that any regular function on $\Bbb{P}^n - (H_i \cap H_j)$ is constant. Now I think I have the proof in my hands which is the…
user38268
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Showing the $V(\mathfrak{a})$ give us a topology on Proj$S$

I'm a bit confused about the proof of Lemma 2.4 on page 76 of Hartshorne's Algebraic Geometry: Lemma 2.4 (a) If $\mathfrak{a}$ and $\mathfrak{b}$ are homogeneous ideals in $S$, then $V(\mathfrak{a}\mathfrak{b})=V(\mathfrak{a})\cup…
porkramen
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Can a connected finite etale cover of a curve over a DVR have a disconnected special fiber?

Let $R$ be a discrete valuation ring. Let $X\rightarrow\text{Spec }R$ be a smooth morphism with geometrically connected fibers of dimension 1. I'm happy to assume that $X$ is the complement of a normal crossings divisor inside a smooth projective…
stupid_question_bot
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conditions birational morphism is isomorphism

Let $X$ and $Y$ be two varieties, and $f:X\rightarrow Y$ be a morphism. Suppose moreover there is a point $Q\in Y$ and $P=f^{-1}(Q)\in X$, such that the restriction $f:X\setminus P\rightarrow Y\setminus Q$ is an isomorphism. So in particular $f$ is…
Alies
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