Mathematics Stack Exchange
Mathematics Stack Exchange

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.

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Blowing up a line $L\subset \mathbb A^3$

I am trying to understand blow-ups, so I'd like to get some advises. Let me show you how I started to blow up a line $L\subset \mathbb A^3$. Any hint/comment/correction is highly appreciated. Suppose $L$ is given parametrically by $x=at,y=bt,z=ct$,…
Brenin
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Line bundles on open subset of projective variety that don't extend over entire variety

I'm looking for an example of the following. Let $X$ be a smooth quasiprojective variety over $\mathbb{C}$ and let $\overline{X}$ be a compactification of $X$. We then have a map $Pic(\overline{X}) \rightarrow Pic(X)$. I want an example where…
user9496
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$f_L:X\to\mathbb P^r$ if finite $\Longleftrightarrow L$ is...?

Suppose we have a line bundle $L$ on an algebraic variety $X$. Let us assume $L$ is globally generated, and let $f_L:X\to \mathbb P^r$ denote the corresponding morphism. Question. Under which condition(s) on $L$ is the morphism $f_L$ finite? For…
Brenin
  • 14,072
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noether normalization theorem geometric meaning

Let $X\subset \mathbb{A}^n$ be an affine variety, let $I(X)=\{f\in k[X_1,\ldots,X_n]:f(P)=0,\ \forall P \in X\}$. We consider the ring $$A=k[a_1,\ldots,a_n]=\frac{k[X_1,\ldots,X_n]}{I(X)}$$ where $a_i=X_i \mod I(X)$. Noether normalization says that…
Federica Maggioni
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Definitions of "linearly normal" variety

I wish I understood the explanation of linear normality on Wikipedia a bit better, and it seems there's actually a mistake at one point. The variety $V$ in its projective embedding is projectively normal if its homogeneous coordinate ring $R$ is…
John Baez
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What happens if we blowup $\mathbb{P}^2$ at more than 9 points?

For $s\leq 8$, if we blowup $\mathbb{P}^2$ at $s$ general points, we get a Del Pezzo surface. I am wondering what happens if $s\geq 9$? How does this 8 being calculated?
minimax
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Cohomology of constant sheaf Z on circle S1

Consider sheaf cohomology defined by derived functor, which means use injective resolutions of sheaves to define cohomology group. Now, if Z is the constant sheaf of group of integers on unit circle S1 with its usual topology. How to computer the…
Rick Sanchez
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The Picard group of projective variety

If $X$ is a smooth irreducible projective variety and its Picard group is $0$, can we conclude that $X$ is a point? (For example, when $X=\mathbb P^n$, then Pic($\mathbb P^n$)=$\mathbb Z$ unless $n=0$)
Summer
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Example II.6.5.2 in Hartshorne (Part I)

Let $k$ be a field and let $A=k[x,y,z]/(xy-z^2)$. Let $X=\operatorname{Spec}A$. What exactly do we mean by a ruling of the cone $Y:y=z=0$? Why is $Y$ a prime divisor of $X$? Edit: What do we mean when we say that $Y$ can be cut-out set-theoretically…
Manos
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Quadrics are birational to projective space

Show that every (irreducible) quadric in $\mathbb{P}^n$ is birational to $\mathbb{P}^{n-1}?$ It is easy to work on examples, like $xt-yz=0$ in $\mathbb{P}^3$ where we first project it to $\mathbb{P}^2$ from $[0:0:0:1]$ i.e. $[x:y:z:t] \mapsto…
Ehsan M. Kermani
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Blow-up and exceptional divisor

I am trying to compute the exceptional divisor of the affine singularity $x^2 + y^2 + z^4 = 0$. First, I take the chart $U_1$ with coordinates $x,y_1,z_1$ verifying $ y = y_1x, z = z_1x$. The equation is $x^2(1 + y_1^2 + z_1^4x) = 0$, so the strict…
user378546
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Fibre of a sheaf of $\mathcal{O}_X$-modules.

Let $\mathcal{F}$ be a locally free sheaf of modules of rank $n$ over some complex manifold $X$. To $\mathcal{F}$ we can associate a vector bundle call, $\pi: V \to X$. The fiber of the sheaf $\mathcal{F}$ is $\mathcal{F}(x) := \mathcal{F}_x…
user7090
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Regular in codimension one VS Singular locus is codimension at least two

In Hartshorne, a scheme is regular in codimension one if the local ring at any (non-closed) point representing a codimension one subscheme is a regular local ring (of Krull dimension one). For varieties, the most naive notion of being regular in…
Kenny Wong
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Hilbert Scheme of Points of Riemann Sphere

I'm looking for a comprehensive note/paper/chapter of a book which discusses the Hilbert Scheme of Points of Riemann sphere ($\mathbb{P}_{\mathbb{C}}^1$) (maybe via a less abstract, more constructive approach?) For the case of $\mathbb{C}^2$ the…
Hamed
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Scheme-theoretic proof of a correspondence between multiple tangents on a plane curve and ordinary $r$-fold points on the dual curve?

Let $X$ be a curve of degree $d$ in $\mathbb{P}^2_k$ where $k$ is an algebraically closed field of characteristic $0.$ We say that a line of $\mathbb{P}^2_k$ is a multiple tangent of $X$ if it is tangent to $X$ at more than one point. If $L$ is a…
Dedalus
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