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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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The morphism of rings inducing the diagonal morphism.

How to see that the diagonal morphism for affine schemes $\Delta: Spec(R) \rightarrow Spec(R \otimes R)$ is induced by the morphism of rings $R \otimes R \rightarrow R$, $r \otimes r' \rightarrow r \cdot r'$? I know this is elementary, but I'm…
karl_christ
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Computing an integral basis of an algebraic function field, $y^4-2zy^2+z^2-z^4-z^3=0$.

I am trying to compute an integral basis for the algebraic extension $K(z,y)$ of $K(z)$ by $y$, with $f(z,y)=0$, $$ f(z,y) = y^4-2zy^2+z^2-z^4-z^3 = 0. $$ $K$ here is either $\mathbb{Q}$ or $\mathbb{C}$, as convenient. I am not really comfortable…
Kirill
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Pull-back of injective morphism of locally free sheaves is injective?

Let $i:X \hookrightarrow \mathbb{P}^n$ be a smooth projective variety. Let $f:\mathcal{F}_1 \to \mathcal{F}_2$ be an injective morphism of $\mathcal{O}_{\mathbb{P}^n}$-modules that are locally free. Is the induced morphism $i^*:i^*\mathcal{F}_1 \to…
Chen
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Why is the restriction map $\mathcal{O}_X(U) \to \mathcal{O}_X(V)$ a flat morphism?

I am reading page 255 of Qing Liu and he claims that if $U,V$ are affine open subsets of a scheme $X$, then $\mathcal{O}_X(U) \to \mathcal{O}_X(V)$ is a flat morphism. Why is this necessarily the case? There are no finiteness assumptions or anything…
user38268
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What does $|mK_V|=|M|+F$ mean for linear system $|mK_V|,|M|$?

I apologize for the ambiguity of the title, the question comes from M.Reid's Young persons guide to canonical singularities (p.355), and I saw similar constructions in many other places. In fact this question is more than explanations of…
Li Yutong
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Birationality and isomorphism of Hirzebruch surfaces

Question 1: How does one show that the $n$-th Hirzebruch surface $$\mathcal{H}_n:=V(u^nX-v^nY)\subset \mathbb{P}^2\times \mathbb{P}^1$$ ($[u:v]\in \mathbb{P}^1, [X:Y:Z]\in \mathbb{P}^2$) and the projective product $\mathbb{P}^1\times \mathbb{P}^1$…
ff90
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automorphisms of rigidified line bundles

Let $\mathcal{L}$ be a line bundle over a proper variety $X/k$. Choose a $k$-rational point $P$ in some fibre of $\mathcal{L}$. Why are there no non-trivial automorphisms of $\mathcal{L}$ fixing $P$? Does this have something to do with…
user5262
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Rational points in the field extension

I am reading Mumford's The Red Book of Varieties and Schemes In Section 4 of Chapter 2, Let $X_0$ be a prescheme over a field $k_0$, and $k$ is a field extension of $k_0$. The prescheme $X$ over $k$ is defined to be $X_0 \times_{\mathrm{Spec}k_0}…
sunkist
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Why is the mobile part of a complete linear system big and nef?

I am trying to learn about linear systems of divisors. Let $X$ be a smooth projective complex surface and $L$ a line bundle. Let's write the complete linear system $|L|=F+|M|$ decomposed in its fixed and mobile part. I don't see why is $M$ big and…
Heitor Fontana
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Are the algebraic-valued points of a variety dense in complex-valued points?

Let $X$ be a variety (integral scheme of finite type) over $\overline{\mathbb Q}$. We may endow the sets $X(\overline{\mathbb Q})$ and $X(\mathbb C)$ of $\overline{\mathbb Q}$- resp. $\mathbb C$-valued points of $X$ with the topologies induced by…
boxdot
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Projective equivalence of two sets of $n+3$ points in $\Bbb P^n$ and on the rational normal curves through each of them

We know that through any $n+3$ points in general position in $\mathbb P^n$ ($n$-dimensional projective space) there is a unique rational normal curve. Let $p_1,p_2,\ldots,p_{n+3}$ be such points and let $q_i$ be their images on the unique rational…
anonymous
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For $f:\Bbb P^1\to\Bbb P^1$ by $[x:y]\mapsto [x^2:y^2]$, show $f_*\mathcal{O}_{\Bbb P^1}\cong \mathcal{O}\oplus\mathcal{O}(-1)$

I am currently preparing for an exam in algebraic geometry and came across the following exercise: Let $k$ be an algebraically closed field. Let $f\colon\mathbb{P}_k^1\to \mathbb{P}_k^1$, $[x,y]\to [x^2,y^2]$. Show that…
Womm
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Stratifications by smooth subvarieties

Let $X$ be an algebraic variety over an algebraically closed field $k$. Then $X$ is said to have a stratification if one can find irreducible locally closed subsets $X_i\subset X$ such that $X=\coprod X_i$ and whenever $\overline X_i$ intersects…
Brenin
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Understanding "only if" statement of Valuative Criteria for Separatedness

This is a follow up question to my earlier question here. I am reading the same document, namely the one by Brian Osserman available here. Now in the document he has stated the Valuative Criteria for Separatedness: Now I am trying to understand…
user38268
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About the Stein factorisation of a morphism defined by a complete linear system on a surface, whose image is a curve

Let $X$ be an algebraic complex projective surface, and let $D$ be an effective divisor on $X$ with empty base locus. Assume that the morphism $\varphi = \varphi_D : X \to \mathbb P^{h^0(D)-1}$ has image a curve $C$ (a sufficient condition is…
Francesco Genovese
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