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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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what is a "section" exactly?

I have read 4 chapters of Hartshorne's Algebraic Geometry, when I go back to the beginning of scheme and the definition of a section, I am kind of confused why we call such a element "section". Let me quote the definition: If $\mathcal{F}$ is a…
finiteness
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Conormal sheaf and ideal sheaf both isomorphic to O(-D)?

Let $i: D\hookrightarrow X$ be a closed subscheme which we can think of as an effective Cartier divisor. Then Hartshorne 6.18 claims that $\mathscr{J}_D\cong \mathscr{O}(-D)$, where the LHS is the ideal sheaf of the closed embedding…
Juan Carlos Ortiz
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The property of line bundle corresponding to birational map

For a variety, if it has a base point free line bundle, then one can define a morphism from the variety to a $\mathbb{P}^n$. And if a variety is a projective variety(in the sense of Hartshorne), it is equivalent to have a very ample line bundle. In…
Li Zhan
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What happens to a variety defined over $\mathbb{C}$ if instead consider the equations over $\mathbb{F}_p$?

Suppose I have $R$ polynomial equations $F_1, ..., F_R$ and say they all have integer coefficients. Let us denote $V_{\mathbb{C}}$ to be the affine variety defined by these polynomials over $\mathbb{C}$. Let $\mathbb{F}_p$ be the finite field of…
Johnny T.
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inverse limit realized as global section

Does the construction (realization) below have some standard name? (Something I could search for?) Does it have any use? Suppose $I$ a directed set, and $(A_\alpha,\pi_{\alpha,\beta} \text{ for } \alpha > \beta \in I $) an inverse system of modules…
gmark
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Do All Conic Sections come with a natural Arithmetic?

I was looking to classify non trivial, commutative, associative, group structures on $\mathbb{R}$ (minus a countable number of point) starting with the trivial ones $$ (\mathbb{R} , + )$$ $$ (\mathbb{R} - \lbrace 0 \rbrace, \times) $$ and then…
Sidharth Ghoshal
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What is the Genus of this Curve? (and why)

I am trying to get an intuitive idea for the genus of an algebraic curve -- this seems like kind of a common question, but I am having trouble building intuition from past answers. I'm looking specifically at this curve: $$ (x^2 + y^2 - 1)(x^2 + y^2…
Andrew Tindall
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The $k$-points of a variety are dense if and only if their images in the base change to $\overline k$ are too, right?

Let $k$ be a field and let $V$ be a variety over $k$ (i.e. a reduced separated finite type $k$-scheme). Let $\overline V$ be the base change $V\times_k \overline k$ to the separable closure $\overline k$ of $k$. We have a canonical fiber product map…
Ben Blum-Smith
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Is the intersection of two quasi-compact open subsets of a scheme quasi-compact?

Is the intersection of two quasi-compact open subsets of a scheme quasi-compact? Is there a counterexample?
Makoto Kato
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Cotangent space of a variety

I encountered this definition of the contangent space of a variety $V$ at $x$, $$T^*_x = \mathfrak m_x/\mathfrak m_x^2 $$ with $\mathfrak m_x = \{f\in k[V]: f(x)=0 \}$. Could you help me with it? From a differential point of view, we can define…
klirk
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Does the category of affine $k$-varieties have finite products?

We use the definitions of this question. Let $\mathrm{Aff}(k)$ be the category of affine $k$-varieties. Let $X, Y$ be objects of $\mathrm{Aff}(k)$. Does the product $X\times Y$ exist in $\mathrm{Aff}(k)$?
Makoto Kato
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Differential operators as a quantization of functions on the cotangent bundle

As the title states, I am trying to see differential operators as a quantization of functions on the cotangent bundle. Specifically, let's assume that $k$ is a field of characteristic $0$, and $X$ is a smooth affine $k$-scheme. Grothendieck gives…
user332239
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Mapping Cartier Divisors to Picard Group

I have some questions about the steps in proof of Prop. 7.1.18 in Liu's "Algebraic Geometry" (page 257): Here we have the Cartier-Divisor $D \in H^0(X,\mathcal{K}^*_X /\mathcal{O}^* _X) = \mathcal{K}^*_X /\mathcal{O}^* _X (X) $ represented by $…
user267839
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Relation between varieties in the sense of Serre's FAC and algebraic schemes

This is a generalization of Hartshorne, Proposition 2.6 and Proposition 4.10, Chapter II. We fix an algebraically closed field $k$. Let $X$ be a topological space. We denote by $\mathcal{F}_X$ the sheaf of $k$-valued functions on $X$. We regard…
Makoto Kato
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Is a morphism between schemes of finite type over a field closed if it induces a closed map between varieties?

This is the converse of this question. Let $X$(resp. $Y$) be a scheme of finite type over a field $k$. Let $f\colon X \rightarrow Y$ be a morphism. Let $X_0$(resp. $Y_0$) be the set of closed points of $X$(resp. $Y$). Then $f$ induces a map…
Makoto Kato
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