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.

29074 questions
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Finding the equations of an image of a rational map

A map from $\mathbb{P^2} \rightarrow \mathbb{P^3}$ is given by: $$f(x:y:z) = (-xyz:xy(x+y+z):xz(x+y+z):yz(x+y+z))$$ How might one find the equations of it's image?
baltazar
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Why is the complement of an affine subset of projective space a hyperplane?

Let $P$ be a projective space of dimension $n$ and $Q$ a linear subspace of it. If the complement of $Q$ is affine, why must $Q$ be of dimension $n - 1$? The following is my thought: Take the homogeneous coordinate system…
Stupid Math Lover
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Basic question about zeros and ideals

These are problems from Miles Reid's book "Undergraduate algebraic geometry" Let $k$ be an algebraically closed field. Question 1. Let $I= (xy,xz,yz) \subset k[x,y,z]$. I want to find $Z(I)$ ok it is clear that $I$ is the union of the three…
user6495
  • 3,957
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A confusion regarding the definition of a quasi-affine variety.

A quasi-affine variety is an open subset of an affine variety. Open under Zariski topology? How does this make sense?
algebraically_speaking
  • 69
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pullback of twisting sheaf

Let $[k]: \mathbf{P}^n \to \mathbf{P}^n, [x_0:\ldots:x_n] \mapsto [x_0^k:\ldots:x_n^k]$ be a morphism. (Why) do we have $[k]^*\mathcal{O}_{\mathbf{P}^n}(1) \cong \mathcal{O}_{\mathbf{P}^n}(k)$?
user5262
  • 1,863
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Variety that is affine and projective is a finite number of points

I was trying to proof the following without any luck. I would appreciate good hints. A projective variety that is isomorphic to an affine variety is a finite number of points.
user181246
  • 41
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algebraic equivalence of line bundles

Two line bundles $L,L'$ on $X/k$ are called algebraically equivalent if $L = M|_{X\times t}$ and $ L'=M|_{X\times t'} $ for some line bundle $M$ on $X \times T$ with $T$ smooth and irreducible and two closed points $t,t' \in T$. Why is this relation…
user5262
  • 1,863
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1 answer

degree of extension of residue fields of cyclic covers

Let $d \geq 2$ be an integer, $k$ a number a field containing $d$-th roots of unity and $X$ and $Y$ smooth varieties over $k$. Let $\pi: Y \to X$ be an unramified cyclic cover of degree $d$. Let $x$ be a point in $X$, with residue field $k(x)$ and…
cyclic93
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What is GAGA for dimension 1 ? (Historical Question)

I know Riemann surfaces are actually algebraic curves, i.e. all Riemann surfaces can be simply embedded into some projective space $\mathbb{P}^n$. But this doesn't indicate me more correspondences between analytic functions and algebraic…
Tei Huang
  • 153
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27 lines on Fermat cubic

Fermat cubic is $S=\{(x:y:z:w)|x^3+y^3+z^3+w^3 =0\} \in \mathbb{P}^3$. It is obvious that 27 lines on Fermat cubic are represented by $(x,ax,z,bz)$ for cube root $a,b$ of $-1$ and their conjugates. But is there any way to find them by resultant? I…
Therefore..
  • 451
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Very ample divisors and the Riemann-Roch theorem

What is the easiest way to prove that a divisor $D$ is very ample if and only if $l(D - P - Q) = l(D) - 2$ for all points $P, Q \in C$. It seems like it might be a consequence of the Riemann-Roch theorem, but I am not sure how to deduce this from…
glebovg
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About Plücker embedding

I'm doing a work about Plücker embedding and I need some help about a few topics. I'm going to list them: $1-$ I know that Plücker embedding is well-defined and is injective. However, Plücker embedding is called an embedding and not an…
Leafar
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Coordinate Ring of Disjoint Union of Affine Varieties

Let X and Y be two varieties in affine n-space such that $X\cap Y=\emptyset$. Let $K[X]=K[X_1,...,X_n]/I(X)$ be the coordinate ring of X. I have managed to convince myself that $X\cup Y$ is an affine variety with an ideal of annihilators $I(X \cup…
user3131035
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Jacobi's criterion for projective schemes?

When can we apply the Jacobi's criterion for the projective variety $V(f_{1}, \ldots, f_{r}) \subset \mathbb{P}^{n}$ in order to find the singularities of the scheme $\mathrm{Proj} \left( k[x_{1}, \ldots, x_{n+1}] / (f_{1}, \ldots, f_{r})…
rla
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image of the abel-jacobi map from a hyperelliptic curve

For a fixed point $x_0\in X$ of a hyperelliptic curve(genus $g$), we can think of the image of Abel-Jacobi map $u: x\mapsto (\int_{x_0}^{x}\omega_1,\ldots,\int_{x_0}^{x}\omega_g)$ into its Jacobian $J(X)$. Then what is it look like? (1) I think its…
user190964
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