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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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Any rational map can be extended to codimension one.

If I understand correctly: Given a rational map $f$, between two (smooth) varieties $X$ and $Y$, with indeterminacy locus $\Sigma$ of codimension 1 in $X$, then $f$ can be extended to a regular map $X\rightarrow Y$. How is this done? Elaboration…
Heitor Fontana
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Rational parametrization of algebraic variety

Suppose I have some algebraic variety (i. e. solution of system of polynomial equations). Sometimes I can find rational parametrization of it - for example, it case of circle defined by $x^2 + y^2 - 1 = 0$ I can parametrize all solutions $x =…
ptashek
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Geometric meaning of intersection multiplicities?

I am wondering about the geometric significance of the intersection multiplicity of two curves as defined in Hartshorne 5.4 (The length of $O_p/(f,g)$ is the intersection multiplicity of $Z(f)$ and $Z(g)$ at $P$). Can someone provide a reference or…
Elle Najt
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Is the composition of blowing-up a blowing-up?

Is the composition of blowing-up of algebraic varieties itself a blowing-up ? I think this is true but I am surprised not to have found any reference, though it seems to be an interesting property. Of course, I'm not able either to prove it…
Lierre
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Why is there no theory of $G$-ic varieties, for linear algebraic groups $G$?

A toric variety is an algebraic variety $X$ with an embedding $T \hookrightarrow X$ of an algebraic torus $T$ as a dense open set, such that $T$ acts on $X$ and the embedding is equivariant. It turns out that, given this setup, essentially all the…
Confused
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$\mathrm{Ext}^n(\mathcal F, i_*\mathcal G) =\mathrm{Ext}^n(i^*\mathcal F,\mathcal G)$?

Given a closed immersion $i: Z \hookrightarrow X$ a coherent sheaf $\mathcal{F}$ on $X$ and a coherent sheaf $\mathcal{G}$ on $Z$, do we have $\mathrm{Ext}^n(\mathcal{F}, i_*\mathcal{G}) = \mathrm{Ext}^n(i^*\mathcal{F}, \mathcal{G})$? For $n = 0$ it…
user5262
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The Gauss Map in Algebraic Geometry

At the moment I am reading Joe Harris' book on algebraic geometry. I am stuck at two points: 1.) The Gauss Map is a regular map 2.) The Gauss Map of a hypersurface is quasi-finite. Both statements can be found on page 188. I have unsuccesfully tried…
user8249
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Hypersurface in $\mathbb P^{n}$ and an exact sequence

Let $X$ be a hypersurface in $\mathbb P^{n}$ defined by the vanishing set of a homogeneous degree $k$ polynomial. Why is the sequence $0 \rightarrow \mathcal O(-k) \rightarrow \mathcal O_{\mathbb P^{n}} \rightarrow i_{*} \mathcal O_{X} \rightarrow…
axel_o
  • 101
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Hartshorne Problem I.3.20

Problem I.3.20 in Hartshorne asks to show that if $Y$ is a variety such that $\dim Y \ge 2$ and $Y$ is normal at a point $P$, then any regular function on $Y-P$ extends to a regular function on $Y$. I am interested in seeing an answer based on the…
Manos
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Epimorphisms of sheaves of sets

Let $X$ be a topological space, and $F$ and $G$ be two sheaves of sets on $X$. Let $\eta : F \rightarrow G$ be a morphism of sheaves. Then how would you show the following: $\eta$ is an epimorphism in the category of sheaves of sets on $X$ if and…
Rankeya
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Exercise from Eisenbud & Harris's The Geometry of Schemes

I've just started learning about schemes, so maybe I'm missing something basic. This is exercise I-24(a): Take Z = Spec$\mathbb{C}[x]$, let $X$ be the result of identifying the two closed points (x) and (x-1) of |Z|, and let $\phi: Z \to X$ be the…
Michael Benfield
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is the empty set an (irreducible) variety?

Is the empty set considered a variety in affine and projective space? By variety, i mean a closed irreducible set in the Zariski topology. On one hand it seems that the empty set satisfies the definition of e.g. an affine variety: it is an…
Manos
  • 25,833
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4 answers

Exercise in Hartshorne

I started reading Hartshorne. Already in the first exercises I stumble across problems. Basically excerise 1.1 ask to prove that $k[x,y]/(y-x^2)$ is isomorphic to a polynomial ring in one variable. Well ok, so I tried the following, define…
wood
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$k$-rational points of a scheme over $K$

Let $K/k$ be an extension of fields, let $X_0$ be a scheme over $k$, and let $X:=X_0\times_k\mathrm{Spec}\;K$, so that $X$ is defined over $k$. In this scenario, I often see the phrase "$k$-rational points of $X$," which confuses me because this…
Jared
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Show that a variety has degree 1 if and only if it is linear

I'm reading Karen Smith's Invitation to Algebraic Geometry and I'm stuck on the following question: Show that a subvariety of $\Bbb{P}^n$ has degree one if and only if it is a linear subvariety. The degree of an $m$-dimensional variety $V \subset…
Mehta
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