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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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Néron-Severi group definition

Let $X$ be a smooth projective variety defined over $\mathbb{C}$. Hartshorne defines the Néron-Severi group as the group of divisors modulo algebraic equivalence. In Lazarsfeld's book "Positivity in algebraic geometry", he defines it is as group of…
user349424
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Question about associated points in Vakil's notes

I have a question about Exercise 5.5.E in Vakil's algebraic geometry notes. Here's the statement: Show that the locus on $\text{Spec } A$ of points $[p]$ where $\mathcal{O}_{\text{Spec } A;[p]} = A_p$ is nonreduced is the closure of those…
Sarah
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For which degrees is being smooth equivalent to being irreducible?

To the best of my knowledge, an algebraic curve of degree $2$ is smooth if and only if it is irreducible. In other words, the only smooth conic sections are the non-degenerate ones. Does this hold for any higher degree algebraic curves? I know…
Chill2Macht
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de Rham cohomology of singular varieties: why completion?

If $X$ is a smooth variety over an algebraically closed field $k$ of characteristic zero one can define algebraic de Rham complex $$ \mathcal{O}_X \to \Omega^1_X \to \ldots \to \Omega_X^n, $$ where $\Omega^i_X = \wedge^i \Omega^1_X$, $n = \dim X$,…
Alex
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Effective divisors exactly those with global sections

Let $X$ be a finite-type scheme over a field $k$. To an effective divisor $D$, there is a global section of the invertible sheaf $\mathcal{O}_X(D)$ (corresponding to the canonical morphism $\mathcal{O}_X \to \mathcal{O}_X(D)$). I've read somewhere…
xuyera
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When a scheme theoretical fiber is reduced?

I'd like to ask some basic things about algebraic geometry. Suppose I have a map $\phi:V\to W$, between affine varieties over $k=\mathbb{C}$. For any point $y \in W$, the scheme theoretical fiber is defined to be $Spec(k[V]\otimes_{k[W]} k )$. The…
Jia-jun Ma
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Exact sequence of sheaves in Beauville's "Complex Algebraic Surfaces"

On the first pages of Beauville's "Complex Algebraic Surfaces", he has a surface $S$ (smooth, projective) and two curves $C$ and $C'$ in $S$. He defines $\mathcal{O}_S(C)$ as the invertible sheaf associated to $C$. I'm assuming that if $C$ is given…
rfauffar
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Line bundle with a nowhere vanishing global section is trivial.

Let $k$ be a field and $X$ be a projective variety over $k$. I think it should be true that if $L$ is a line bundle on $X$ such that exists $s \in \Gamma(X,L)$ with $s_x \neq 0$ for all $x \in X$, then $L$ is isomorphic to $\mathcal{O}_X$. At least…
user110071
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Why we need to add the notion of "separated" to the notion of variety?

In most case, the definition of a variety over a field $k$ at least requires that being "of finite type" and being "separated". It has no question for me that being of finite type, since we always like finite. I donot know the reason why we…
wxu
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Irreducible component of fibre product of schemes

I have some questions about algebraic geometry which might be elementary and boring (sorry). Let $R$ be a ring - say, an integral domain which is Noetherian. Let $X$ and $Y$ be $R$-varieties - that is, integral separated schemes of finite type over…
Evariste
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Geometric interpretation of Noether normalisation and finding a transcendence basis

Take the polynomial $f = x_1 x_2 + x_2 x_3 + x_3 x_1 \in k[x_1,x_2,x_3]$, and define the set $V = Z(f) = \{(x_1,x_2,x_3) \ | \ f(x_1,x_2,x_3) = 0 \} \subset \mathbb A ^3$. Consider the coordinate ring of $V$, given by $k[x_1,x_2,x_3]/(f) \cong…
Matt
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Fourier Mukai transform of skyscraper sheaf through flat kernel.

Let $X,Y $ be smooth projective varieties over a field $k$. The following example is taken from Huybrecht's "Fourier Mukai Transforms in Algebraic Geometry" [example 5.4.vi]: "Suppose $\mathcal{P}$ is a coherent sheaf on $X\times Y$ flat over $X$…
Carsten
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What does this notion of scheme morphism mean?

I apologize for the naivety of this question but I am a beginner in algebraic geometry. Moreover, I realized that the initial question (quoted at the end) was not formulated very clearly. Hopefully, I can do better now involving the things learned…
Daniel Dreiberg
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Definition of homogeneous ideal

I'm a little confused about the definition of a homogeneous ideal. I have the following two definitions: An ideal $I\subset k[X_{0}, \dots, X_{n}]$ is homogeneous if $I$ is generated by (finitely many) homogeneous polynomials. An ideal $I\subset…
guestfromthepast
  • 155
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Spectrum of a quasi-coherent algebra on an algebraic stack

Let $X$ be an algebraic stack and $\mathcal{A}$ a quasi-coherent $\mathcal{O}_X$-algebra. Define the stack $\mathrm{Spec}(\mathcal{A})$ by $\mathrm{Spec}(\mathcal{A})(T) := \{(f,h) : f \in X(T), h \in \hom_{\mathrm{Alg}(\mathcal{O}_T)}(f^*…
Martin Brandenburg
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