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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 hyperelliptic curve is never a complete intersection.

Show that any hyperelliptic curve is never a complete intersection. As any curve of genus greater than 1 is either hyperelliptic or canonical, I think we can equivalently show that any curve of genus greater than 1 which is a complete intersection…
Alex Fok
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Structure Sheaf on Scheme

Given a scheme $X$, its structure sheaf on the elements of the cover by affines is pretty easy to define, say $\mathcal{O}_X(Spec(A))=A$. But how is it then defined on the union of two of these sets? In general, it is clear how to define open…
Jonathan Beardsley
  • 4,503
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any two sets of $n+2$ points are projectively equivalent in $\mathbb{P}^n$

Problem: Any two sets of $n+2$ points in general position in $\mathbb{P}^n$ are projectively equivalent. In thinking about this problem, it is natural for me to reduce it to linear algebra considerations. So i have $n+2$ lines in $k^{n+1}$ spanned…
Manos
  • 25,833
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Surfaces in $\mathbb P^3$ not containing any line

Let $d \geq 4$. I'm interested by know if there is a surface $S$ of degree $d$ in $\mathbb P^3_{\mathbb C}$ such that $S$ does not contains a line. I know I have no idea how to do it.
user139243980
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Projective Nullstellensatz

I'm confused about the proof of the Nullstellensatz for projective varieties. If $J \subset k[x_0, \ldots , x_n]$ is a homogeneous ideal, we may regard $V(J)$ as a closed subset $ V(J) = V \subset \mathbb P_k^n$ and also as a closed subset $V(J) =…
Matt
  • 71
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Example of different schemes with the same space and stalks

What is an example of two non-isomorphic schemes $(X,\mathcal{O}_X)$ and $(X,\mathcal{O}_X')$ with the same topological space such that there are isomorphisms $\mathcal{O}_{X,p}\cong \mathcal{O}_{X,p}'$ of all stalks?
Alex
  • 221
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When does a generic point map to a generic point?

This question may be too vague, so feel free to specialize to particular examples. Given a morphism of schemes $f:X\to Y$, I want to know what conditions one can impose on $f,X$ or $Y$ such that a generic point of $X$ will map to a generic point of…
Tian An
  • 817
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Proof that all rational maps $\mathbb{P}^1\to\mathbb{P}^N$ are regular without using codimension?

There is a theorem in Shafarevich's Basic Algebraic Geometry (Theorem 3, pg. 109) which states that if $X$ is a nonsingular variety, and $\varphi\colon X\to\mathbb{P}^N$ a rational map to projective space, then the set of points at which $\varphi$…
Kaitlyn Schweed
  • 229
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Not every complex manifold is of the form $X_{an}$ where $X$ is some algebraic variety

I want to show that not every complex manifold is of the form $X_{an}$ where $X$ is some algebraic variety, providing a counterexample. The candidate for this counterexample seems to be the open unit disk $D$ in $\mathbb{C}$. But how can I proof…
TheWanderer
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Question about the construction of Mukai flop

Let X be a symplectic complex manifold of dimension $2n $, i.e. there exists a non degenerate holomorphic 2-form $\sigma $ such that $ H^0 (X,\Omega^2)=\mathbb {C}\cdot\sigma $. Suppose that there exists a submanifold $ P\subset X $ such that…
User3773
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Proving the Nullstellensatz for homogeneous ideals

I'd like to prove the following: If $\mathfrak{a} \subseteq k[x_0, \ldots, x_n]$ is a homogeneous ideal, and if $f \in k[x_0,\ldots,x_n]$ is a homogeneous polynomial with $\mathrm{deg} \ f > 0$, such that $f(P) = 0 $ for all $P \in Z(\mathfrak{a})$…
Matt
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Is the set of complex solutions to $x^2+y^2 = 1$ isomorphic to $\mathbb{C}^*$?

In his article about Grothendieck, Edward Frenkel states that the set of complex solutions to the equation $x^2+y^2 = 1$ is "a plane with one point removed." I'm curious how this can be made precise. Since the plane minus the origin is…
lie341
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Proof of Nike's trick: Two affine open subsets contain a simultaneously distinguished open subset

I'm trying to work through this proof of Nike's tick. Statement of the lemma: Let $ U_{i} = Spec\ A_{i} $ for $ i\in\{1,2\} $ be two open affine subschemes of a scheme $ X $. For $ x\in U_{1}\cap U_{2} $ there exists an open affine subscheme $V $…
PeterM
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Monomorphisms of sheaves gives an injection of stalks

How do I show that a monomorphism $F \rightarrow G$ of sheaves induces an injection on stalks? When showing that monomorphism is an injection on sets one uses the maps $x \mapsto a$ and $x \mapsto b$ where $a,b$ are some fix elements such that they…
erwin
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1 answer

Equivalence of two definitions of ramification index

Let $\phi:Y\to X$ be a morphism of $\Bbbk$-varieties. In Görtz and Wedhorn's Algebraic Geometry 1, the ramification index is defined by $$e_Q(Y) := \mathrm{length}_{\mathcal{O}_{Y,Q}}\left(\mathcal{O}_{Y,Q}\right)$$ and then $e_{Q/P} := e_Q(Y_P)$…
Jesko Hüttenhain
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