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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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Description of inverse image ideal sheaf as kernel of a morphism

Let $Y \rightarrow \mathbb{P}^n$ be a finite scheme and let $Z \rightarrow \mathbb{P}^n$ be a closed subscheme. I read that there is an exact sequence $0 \rightarrow \mathcal{I}_Z \cdot \mathcal{O}_Y \rightarrow \mathcal{O}_Y \rightarrow…
I like Cake
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Quotient Scheme of a Proper Scheme

Let $X$ be a scheme proper over $\mathbb{Z}$, $G$ a finite group acting on $X$. Suppose that the quotient scheme $Y := X/G$ is well-defined. Should $Y$ be then proper over $\mathbb{Z}$ as well?
OldStudentL
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Exercise 8.1.A of Vakil's FOAG

In Vakil's FOAG, definition 8.1.1 reads as A morphism $\pi : X \to Y$ of schemes is a closed embedding if $\pi$ is affine, i.e., for every affine open subset $\mathrm{Spec} B$ of $Y$, $\pi^{-1}\mathrm{Spec} B \cong \mathrm{Spec} A$ is affine…
Yu Ning
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Determining the coefficients of divisors under a blowup of smooth varieties along smooth subvarieties.

Let $ \pi: Y \rightarrow X $ blowup of a smooth variety along a smooth subvariety, with exceptional divisor $ E. $ Then $$ \operatorname{Pic}Y \cong \pi^{*}\operatorname{Pic}X \oplus \mathbb{Z}E $$ (can I find a proof of this fact in most books?…
AG Novice
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Higher order taylor expansions in algebraic geometry for regular rings

Let $A$ be a local algebra over a field $k$ with maximal ideal $m,$ such that $A/m=k$. Suppose that $A$ is regular. Then it seems to me that we can think of the tangent space as being any of the equivalent notions given by…
xlord
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The fibers of a map to $\mathbb{P}^1$

Suppose $X$ is a surface and $D$ is a prime divisor such that $|D|$ gives rise to a surjective morphism $f:X\to \mathbb{P}^1$. Is every fiber linearly equivalent to $D$? My thought process was that we can find one point $p\in \mathbb{P}^1$ whose…
user64480
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Shape of zero set of homogeneous polynomial

Let $f$ be a homogeneous polynomial in $d$ variables of degree $n$ over the real numbers. What does its zero set $V(f)$ look like? Is it a "hypersurface"? Is it connected in the metric topology of $\mathbb{R}^d$?
Manos
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Definition of open set in Zariski Topology

I'm being confused by the definition of the open set in Shafarevich's Basic Algebraic Geometry I. He says that in p.23: "Let $X\subset A^n$ be a closed subset of affine space. We say that $U\subset X$ is open if its complement $X∖U$ is closed." how…
Ajat Adriansyah
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Union of all hyperplanes in projective space

I recently read the paper "Compactification of a Drinfeld Period Domain over a Finite Field" by Pink and Schieder. (link to the paper) I am confused about two statements appearing in this paper: (1) removing all proper $\mathbb{F}_q$-rational…
AAAS
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A function vanishing nowhere in the coordinate ring of an affine variety is invertible

This is exercise 4.2.1 of Smith's "Algebraic Geometry" - I have tried to look for a previous question, but cannot find any (probably because it is very easy, yet I am stuck on what I think is the final step). The setup: $X$ is an affine variety, $f$…
mi.f.zh
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Do these definitions of irreducibility of algebraic sets coincide?

I am reading the book Numerically solving polynomials systems with Bertini in which they define a manifold point $p^* = (p_1^*,\ldots,p_m^*)$ of an algebraic set $X$ to be a point in $X$ with an open neighborhood $U\subset X$ such that for some…
Jarne Renders
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"Coordinate ring'' of the algebraic $n$-torus

I'm trying to calculate the "coordinate ring'' of the algebraic $n$-torus $(\mathbb C^n)^\ast$: If $X:=\mathbb C^n$ and $U:=(\mathbb C^n)^\ast$, we have that $X$ is an irreducible affine algebraic set ad $U$ is an open subset (with the Zariski…
Dubious
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A rational map is not a map,...Why???

Def of rational map Let X and Y be varieties. A rational map $\phi:X \to Y$ is an equivalence class of pairs $(U,\phi_U)$ where $U$ is a nonempty open subset of X, $\phi_U$ is a mophism of U to Y and where $(U,\phi_U)$ and $(V,\phi_V)$ are…
Jean Billie
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Bertini's theorem for singular varieties over an algebraically closed field

Let $k$ be an algebraically closed field (of arbitrary characteristic), $X$ a projective $k$-variety (= integral projective $k$-scheme) and $f: X \to P := \mathbb{P}_k^n$ a closed immersion. (And identify the set of hyperplanes of $P$ with $P' :=…
k.j.
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Birational smooth curves

Let $X$ and $Y$ be connected schemes that are smooth and proper of relative dimension 1 over $\mathbb{Z}[1/n]$ for some positive integer $n$. If the function fields of $X$ and $Y$ are isomorphic, are $X$ and $Y$ isomorphic themselves?
user700841
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