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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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How to show $\alpha_d : M_d \to \Gamma(X, \widetilde{M(d)})$ an isomorphism for sufficiently large $d$?

Let $S$ be a (positively) graded ring and $X = \operatorname{Proj} S$. Suppose $S$ is generated by $S_1$ as an $S_0$ - algebra and suppose further that $S_1$ is a finitely generated $S_0$ - module. Thus there are $x_1,\ldots,x_r \in S_1$ such that…
user38268
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subscheme of blow up

Let $X$ be a noetherian scheme, $Y$ a closed subscheme defined by ideal sheaf $\mathcal{I}$ and $\pi : \widetilde{X} \rightarrow X$ be the blowing up of $X$ along $Y$. Let $Y'$ be the subscheme of $\widetilde{X}$ defined by the ideal sheaf…
Sang Cheol Lee
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Is the restriction of a flat morphism flat?

Let $X$ be a projective scheme, $X_1, X_2$ be closed subschemes of $X$. Let $f:X \to S$ be a flat morphism for some scheme $S$. Denote by $i_1$ and $i_2$ the natural inclusion maps from $X_1$ and $X_2$, respectively to $X$. Assume that the…
Chen
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Is $\mathcal{Z}(xy-z^2)$ isomorphic to $\mathbb{A}^2$ for a not algebraically closed field?

Context: This is exercise $24$ from Dummit and Foote's section on Algebraic Geometry. Let $V=\mathcal{Z}(xy-z^2)\subset \mathbb{A}^3$. Where we work over the base field $k$. If $k$ is a finite field, then $V\simeq W \Longleftrightarrow …
RyeCatcher
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A generalized criteria of being projective spaces?

Here is an exercise V.1.11.12.1 in J. Kollar's book Rational curves on algebraic varieties: Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$. Let $\mathscr{L}$ be a line bundle on $X$ such that…
WakeUp-X.Liu
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Properties of canonical divisor of fibres

Suppose $f: X \to Y$ is a morphism between varieties, let $y \in Y$ be a closed point, and $X_y$ be the fibre of the morphism over $y$. I learned that the canonical divisor $K_X$ of $X$, the canonical divisor $K_{X_y}$ of $X_y$, and the relative…
Li Yutong
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How to show that a discrete valuation ring has only two prime ideals?

In the book Algebraic Geometry by Hartshorne, page 74, it is said that the spectrum of a discrete valuation ring $R$ has only two points. How to show that a discrete valuation ring has only two prime ideals? In…
LJR
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Why $\operatorname{Proj}\mathbb{Z}[x,y,z]/(y^2z+yz^2-x^3+xz^2)$ is fibered surface over $\operatorname{Spec}\mathbb{Z}$?

I am reading the Liu's Algebraic Geometry and arithmetic curves, p.348, Example 3.2. and stuck at some point. Definition 3.1 ( In his book p.347 ). Let $S$ be a Dedekind scheme. We call an integral, projective, flat $S$-scheme $\pi : X \to S$ of…
Plantation
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On a Geometric Interpretation of the Local Criterion for Flatness in Eisenbud's

The local criterion for flatness goes this way: Let $\phi : (A,m)\rightarrow (B,m')$ be a local morphism of local Noetherian rings, and $M$ a finitely generated $B$-module. If $x\in m$ is a non-zero divisor on $A$ then $M$ is flat over $A$ iff…
brunoh
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Injectivity of map on sheaves $f^\sharp : \mathcal{O}_X \to f_*\mathcal{O}_Y$. Hartshorne $2.18$.

Let $\varphi : A \to B$ be a homomorphism of rings, and let $f: Y = \operatorname{Spec} B \to X = \operatorname{Spec} A$ be the induced morphism of affine schemes. Show that $\varphi$ is injective if and only if the map of sheaves $f^\sharp :…
Laura
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how to define the composition of two dominant rational maps?

Let's follow Hartshorne's definition, according to which, a rational map $\phi:X \rightarrow Y$ from variety $X$ to variety $Y$ is an equivalence class $\langle U,\phi_U \rangle$, where $U$ is open in $X$ and $\phi_U: U \rightarrow Y$ is a morphism…
Manos
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Local coefficient systems for schemes

Let $B$ be a path connected topological space with universal cover $p:X \rightarrow B$. Let $G = \pi_1B$, then for a $G$-module $M$ we can form $X \times_G M = X \times M / (g x,m) \sim (x, g m)$ where $M$ is given the discrete topology. The…
Al G. Bra
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Compact variety which is not projective

While reading Andreas Gathmann's notes on Algebraic Geometry, I stumbled upon this statement: "Projective varieties form a large class of “compact” varieties that do admit such a unified global description. In fact, the class of projective varieties…
user39280
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Cohomology exists if and only if its forced to

When setting up the foundations of sheaf cohomology, one proves that flasque sheaves have no cohomology. The proof in Hartshorne is an induction, and relies on two properties of flasque sheaves: if $0 \to\mathcal{F}'\to\mathcal{F}\to\mathcal{F}''\to…
user960774
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Surjectvity of multiplication map $H^0(X,L)^{\otimes t} \to H^0(X,L^{\otimes t})$

Given a proper, integral scheme $X$ over an arbitrary field and an invertible sheaf $L$ on $X$, does there exist any integer $k \gt 0$ such that the maps $H^0(X,L^{\otimes k})^{\otimes t} \to H^0(X,L^{\otimes kt})$ become surjective for all integers…
Rührei
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