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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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What are some good examples of (non-quasicoherent) sheaves not satisfying the conclusion of Hartshorne Lemma II.5.3?

Hartshorne, Algebraic Geometry, Lemma II.5.3 reads (roughly): Let $X = \operatorname{Spec} A$, let $f \in A$, and let $\mathscr{F}$ be a quasicoherent sheaf on $X$. (a) If $s \in \Gamma(X, \mathscr{F})$ with $s|_{D(f)} = 0$ then $f^n s = 0$ for $n…
Daniel McLaury
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Lefschetz Hyperplane Theorem for Picard groups of surfaces?

Griffiths and Harris, On the Noether-Lefschetz Theorem and Some Remarks on Codimension-Two Cycles, Math. Ann. 271, 31-51 (1985), states [...] look at the restriction $$r_1 : \operatorname{Pic}({\widetilde{S}}) \to…
Daniel McLaury
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1 answer

Doubt in a proof on projective varieties from Hartshorne

In Algebraic Geometry by Hartshorne in the proof of theorem 3.4 in Chapter 1 he gives an isomorphism of $k[y_1,...,y_n]$ with $k[x_0,...,x_n]_{(x_i)}$ by sending $f(y_1,...,y_n)$ to $f(x_0/x_i,...,x_n/x_i)$ leaving out $x_i/x_i$. Then he says that…
R_D
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2 answers

How to find a bijection between R-valued points of X and local ring homomorphisms?

I am trying to prove the following fact for a homework assignment in algebraic geometry: Let R be a local ring, and X a prescheme. Show that there is a one-one correspondence between R-valued points on X and pairs (x, g) of points x on X and local…
Gabe Cunningham
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Defining the map between tangent space in locally ringed space

I had a doubt studying locally ringed space about what is the canonical map between tangent spaces in the case the residue field is different: Let $(f,f^*):(X,O_X) \to (Y,O_Y)$ a morphism of locally ringed space. Then is clear that given $x \in X$…
Tuc
  • 71
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Projective curve $x^d+y^d+z^d=0$ is nonsingular using Jacobian matrix

According to this question: Nonsingular projective variety of degree $d$, the curve $x^d+y^d+z^d=0$ in $\mathbb{P}^2$ is nonsingular. I'm trying to prove this. Hartshorne defines nonsingular points using the Jacobian matrix in $\mathbb{A}^n$ and…
PeterM
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1 answer

Global sections on non-reduced proejctive schemes

Suppose $X$ is a non-reduced finite type seperated projective scheme over a field $k$, can it happen that $\Gamma(X,O_X)=k$?
user93417
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Blowup of cone over Veronese surface

Consider the affine cone $X \subset \mathbb A^6$ over the Veronese surface $V \cong \mathbb P^2 \subset \mathbb P^5$. After blowing up the cone point we get a resolution $Y \rightarrow X$, with exceptional divisor $E \cong \mathbb P^2$. How do you…
Davide Cesare Veniani
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Two questions on the definition of $\mathcal{O}_X(U)$ for an affine scheme $X$.

Let $X=\operatorname{Spec}(A)$ be an affine scheme. Hartshorne defines $$ \mathcal{O}_X(U)=\{s\colon U\to\coprod_{\mathfrak{p}\in U} A_\mathfrak{p} \mid s(\mathfrak{p})\in A_\mathfrak{p} \text{ and } s \text{ is locally a quotient of elements of…
user8463524
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What is $k(X)[Y]$ and why is it a principal ideal domain? From a proof in Fulton's Algebraic Curves

Fulton's "Algebraic Curves" says the following: Let $F$ and $G$ be polynomials belonging to $k[X,Y]$, where $k$ is a field. Let $F$ and $G$ not have a single common factor in $k[X][Y]$. Then they do not have a common factor in $k(X)[Y]$ either. As…
freebird
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Torsion and Coherent Sheaves

Let $X$ be a smooth curve defined over a field and $F$ a coherent sheaf on $X$. I would like to show that $F/F_{t}$ is locally free, for $F_{t}$ the torsion subsheaf of $F$. Since $F$ is coherent it is enough to show the stalk $F_{p}$ is free as an…
Dreyfus14
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Coordinate ring of the product of projective variety

Let $X\subseteq \mathbb{P}^r,Y \subseteq \mathbb{P}^s$ be two projectve varieties,what is the coordinate ring of $X\times Y$(segre embedding)?Is it true that $$S(X\times Y)=S(X)\otimes_k S(Y)?$$ I also want to know what is the dimension of $ X\times…
Wei Xia
  • 892
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List of exercises and examples to see the geometry behind algebraic geometry

What exercises should one solve (understanding proofs included) to gain an intuition for algebraic geometry? What are examples of (not too hard) problems that algebraic geometry handles easier than elementary approaches?
Adiji
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Chow group of zero cycles of a product

I have been starting to learn about about chow groups. I don't know much yet, so hopefully the following is trivial: :-) For a smooth (projective, if you like) variety $X$ over a field $k$ I will write $CH_n(X)$ for the Chow group of $n$-cycles up…
Tom Bachmann
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1 answer

Hartshorne exercise 1.3.8

I'm trying to solve exercise 1.3.8 from Hartshorne's Algebraic Geometry: Let $ H_{i} $ and $ H_{j} $ be the hyperplanes in $ \mathbb{P}^n $ defined by $ x_i = 0 $ and $ x_j = 0 $ with $ i \neq j $. I Show that any regular function on $\Bbb{P}^n -…
PeterM
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