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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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Automorphisms of Surfaces and Quotients

Let $X$ be a surface (algebraic projective smooth complex) and suppose $\sigma$ is an automorphism of finite order $d$. Let $Y=X/\sigma$. I wonder under which simple conditions on $\sigma$ is $Y$ a smooth surface. For example it seems reasonable…
Heitor Fontana
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Affine variety isomorphism between $\mathbb{V}(y-x^2)$ and $\mathbb{V}(y-x^3)$

I am asked to prove that $\mathbb{V}(y-x^2)$ and $\mathbb{V}(y-x^3)$ are isomorphic, but I cannot find an inversible morphism from $\mathbb{V}(y-x^2)$ to $\mathbb{V}(y-x^3)$. In order to make the morphism inversible, I think we can only consider…
hxhxhx88
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Two basic questions about uniformizers in algebraic curves

I'm recently trying to study basics about algebraic curves. However, apparently I'm still quite unfamiliar with the subject as there have occured 2 questions that seem quite basic to me, yet I don't directly see a way to make them clear to…
Louis
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Let $X=\operatorname{Spec} k[w,x,y,z]/(wz−xy)$. Show the Weil divisor cut out by $w=x=0$ is not locally principal.

This is exercise 14.2.R from Vakil's notes. Let $X=\operatorname{Spec} k[w,x,y,z]/(wz-xy)$. Let $Z$ be the Weil divisor cut out by $w=x=0$. I want to show that $n[Z]$ is not locally principal for all $n\neq0$. To do this I assumed $n[Z]$ is locally…
Gazerun
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intersection of two affine open sets of a scheme

In page 299 of Ravi Vakil's lecture "Foundations of algebraic geometry" , there is a statement: For a scheme X, the category of affine open sets, and distinguished inclusions, forms a filtered set. Given two affine open sets U and V of the…
Yubin
  • 247
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Zariski tangent space and $K[\epsilon]/(\epsilon^2)$

I want to prove that the Zariski tangent space at $x\in X$ ($X$ is an affine scheme) is isomorphic to $Hom_K(X,K[\epsilon]/(\epsilon^2))$ (K is the residue field at $x$). I want to say that $$Hom_K(\mathfrak{m}_{X,x}/\mathfrak{m}_{X,x}^2,K)\simeq…
ArthurStuart
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Action of $PGL(n+1) $ on $\mathbb P^n$!

what is the action of $PGL(n+1)$ on Projective space $\mathbb P^n$? (over algebraic closed field)
nahid fardmanesh
  • 97
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Desingularization of ordinary double point of a surface

Let $X: xz=y^2 \subset \mathbb{A}^3$ be a surface with ordinary double point. It is claimed that there exists a resolution $f:Y \to X$ for which the exceptional divisor is a curve $E \cong \mathbb{P}^1, E^2=-2$. I don't know how to show the above…
Li Yutong
  • 4,065
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Extending sheaves to union of curves

Let $C_1$ and $C_2$ be projective curves in $\mathbb{P}^3$. Assume further that $C_1 \cap C_2$ is finitely many points. Let $\mathcal{F}$ (resp. $\mathcal{G}$) be a sheaf defined over $C_1$ (resp. $C_1 \cup C_2$). Assume that…
Chen
  • 981
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Inequalities between chern characters and chern classes

When speaking of a sheaf $E$ on a complex surface $X$, if an author writes $$ch_2(E)
AlgGeomQ
  • 195
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Prove that the Frobenius map $f_p : \mathbb{P}^n \rightarrow \mathbb{P}^n$ is a regular map

I wish to prove that the Frobenius map $f_p : \mathbb{P}^n \rightarrow \mathbb{P}^n$, defined by $[x_0, \dots, x_n] \mapsto [x_0^p, \dots, x_n^p]$, is a regular map but not an isomorphism (for $p \geq 2$, of course).
user96063
  • 31
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Tangent sheaf of a rational curve is a subsheaf of tangent sheaf of ambient space

Let $X$ be a smooth, proper variety, and $f: \mathbb{P}^1 \to X$ be a finite morphism. Then it is claimed that $f^* T_X$ contains $T_{\mathbb{P}^1}$ as a subsheaf. However, I did not see the reason. I think we have exact sequence $$f^* \Omega_{X}…
Li Yutong
  • 4,065
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Homotopy invariance in Fulton's book.

I have a question about a proof in Fulton's Intersection theory. In the first chapter, he wants to prove that Chow groups are invariant with respect to homotopy, and the usual reductions tell us that's it's tantamount to proving it for $X$ an…
Minesotafan
  • 31
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A Question about the local nature of coherent sheaf

In the definition of a coherent sheaf: 1)$\mathcal{F}$ is of finite type; 2)for any open set $U$, any $n$, any $u: \mathcal {O}^n_U \rightarrow \mathcal{F}_U$ has kernal of finite type ; Does it imply that if $(X,\mathcal{F})$, $X$ is covered by…
user93417
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how is a cubic curve topologically equivalent to a torus?

Can someone explain this in simple terms? What I don't get is, how can a 1-D curve be topologically equivalent to a 2-D surface? I guess isomorphism cannot hold between the 2 sets, and I don't see the homeomorphism.
Yan King Yin
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