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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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Is a faithfully flat quasi-compact base change injective?

Let $h\colon S' \rightarrow S$ be a faithfully flat quasi-compact morphism. Let $f, g\colon X \rightarrow Y$ be morphisms of $S$-schemes. Let $X', Y'$, and $f', g'$ be the base changes by $h$. Suppose $f' = g'$. Can we say $f = g$?
Makoto Kato
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canonical bundle of Veronese embedding

Suppose we are given a complete intersection $X$ of codimension $r=n-d$ in $\mathbb{P}^n$ where the degrees of the hypersurfaces are $d_i$ and $d$ is the dimension of $X$. Then the canonical bundle $\omega_X=\mathcal{O}_X(\sum d_i-n-1)$. Now suppose…
user109227
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Checking separatedness after glueing two schemes

Let $X \to S$ and $Y \to S$ be separated morphisms of schemes and suppose that $X$ and $Y$ can be glued over an open $U \subseteq S$, in such a way that each fiber of the new morphism $Z \to S$ is completely contained in $X$, completely contained in…
Evariste
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How close is the analogy between regular systems of parameters and smooth coordinate charts?

Let $X$ be a smooth variety with dimension $ n$ and $ p \in X $ a closed point. By definition, $ \mathscr{O}_p$ is a regular local ring, so we can choose a regular system of parameters $ u_1, \dots, u_n$ to generate the maximal ideal in $…
DBr
  • 4,780
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1 answer

Preservation of separatedness of a scheme of finite type over a field by shrinking the base field

This is a generalization of this question. Let $k$ be a field. Let $k'$ be an extension field of $k$. Let $X$ be a $k$-scheme of finite type. Suppose $X\times_k k'$ is separated over $k'$. Is $X$ separated over $k$? If yes, how do you prove it?
Makoto Kato
  • 42,602
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Why are these line bundles isomorphic? [modified]

Reading a book I met the following claim, and I don't understand how to justify it. [Actually I misunderstood the claim, below is the corrected version of it] Let $X$ be a variety and $E\subset X$ a divisor. Suppose we have a global section $$ s:…
Abramo
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canonical vs dualizing bundle

It is well known that for smooth projective varieties over a field $k$ the dualizing sheaf is isomorphic to the canonical sheaf. My question is now if this holds for the relative case assuming both schemes in question are smooth projective…
user109227
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relative dualizing sheaf

Suppose we are given a projective smooth variety $X$ and a locally free sheaf $\mathcal{E}$. Now consider $\pi:\mathbb{P}(\mathcal{E})\rightarrow X$. One can define the relative dualizing sheaf as it is done for example in Fourier--Mukai transfors…
user109227
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birational map P1

I have the following problem to prove that $V(x_0^3-x_2x_1^2)\subset\mathbb P_k^2 $, where $k$ algebraically closed field, is birational to $\mathbb P_k^1$. I'm a beginner at this stuff, so someone tell me please if I'm following the right track. I…
guest
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A question on the dimension

Let $Y:z=x^{2}+y^{2}$ and $Z:z=-x^{2}-y^{2}$ be two affine varieties in $\mathbb{R}^{3}$, then how does it satisfy the Affine Dimension Theorem here (on Page 3)?
celilia
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What is going on with the map $X \mapsto X^2$

Consider the homomorphism $f : \mathbb{C}[X] \to \mathbb{C}[X]$, $X \mapsto X^2$. It induced a morphism of affine schemes $\operatorname{spec} f : \mathbb{A} \to \mathbb{A}$ which topologically is the identity function. How am I meant to think about…
DBr
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Turning varieties into affine varieties?

Given an affine variety, we can forget it into a variety. The other way round, given a variety $X$ we can look at $$\mathrm{spec}(\mathcal O_X(X)$$ Is this necessarily an affine variety (a priori its just an affien scheme)? If so, am I right that…
Jan
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Birational morphism and etale base change

I have a question which is probably well known. Let $f:X\rightarrow Y$ be a birational morphism of reduced schemes and $Y_1\rightarrow Y$ be an étale morphism. Is it true that $X\times_Y Y_1\rightarrow Y_1$ is birational?
hjia
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Question about proof of "nonsingularity of a point iff the local ring at that point is regular"

In Hartshorne, page 32 Theorem 5.1, the theorem says "Let $Y\subset\mathbb{A}^n$" be an affine variety. Let $P\in Y$ be a point. Then $Y$ is nonsingular at $P$ if and only if the local ring $\mathscr{O}_{P,Y}$ is a regular local ring. Now in the…
Robert Auffarth
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Quotient of quasi-projective variety

Let $X$ be a quasi-projective variety over an algebraicaly closed field and $G$ be a finite group acting on $X$ through automorphisms. Can you tell me how to prove that the quotient $X/G$ is also a quasi-projective variety. In particular: Why is…
philipp
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