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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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Triviality of vector bundles

Let $X$ be a proper curve, not necessarily smooth nor reduced, and $E$ a vector bundle on $X$ of rank $r$. Assume we know that $H^0(X,E)\geq r$ and $H^0(X,E^{\vee})\geq r$, can we conclude that $E$ is trivial? Is there a cohomological criterion to…
jikki
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How can we check the gluing property of sheaf of ideals?

For a ringed space $(X,\mathcal{O}_X)$, one can define a sheaf of ideals $\mathcal{J}$ of $\mathcal{O}_X$. Then how can we see the $\mathcal{J}$ satisfies the conditions of sheaf? Especially, I cannot show the gluing property. For an open set $U$…
User0829
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Space of algebraic curves of degree $d$ is compact

In an article I am reading it states "the space of algebraic curves of a given degree $d$ is compact". It seems to take this as a basic fact, as there is no explanation on this. I was wondering could someone please explain what this means?
Johnny T.
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Is the structure morphism flat?

Let $X$ be a smooth projective variety over $k$. Variety here meaning a separated geometrically integral $k$-scheme of finite type. Is the structure morphism $f : X \to \text{Spec}\;k$ flat? I guess this means we should check that for any $x \in X$…
Ruben
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cohomology class of subvarieties of $\mathbb{P}^n$ in $H^p(\mathbb{P}^n, \Omega^p)$

This is the exercise III 7.4 of Hartshorne. If you don't have this text in your hand, see this post. I'm trying (b): If $X = \mathbb{P}^n$, identifying $H^p(X, \Omega^p)$ with $H^0(X, \Omega^0) = k$, show that $\eta(Y) = \deg Y$, the degree of the…
k.j.
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Image of quasiprojective variety under closed map

Let $f: X\to Y$ be a regular map of projective varieties that is closed (in the sense that it takes Zariski closed sets to Zariski closed sets). Let $V\subset X$ be a quasiprojective subvariety (i.e. locally closed and irreducible). Is $f(V)$ a…
birdiehoweblazer
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How to show $\phi(X)$ contains a non-empty open set of its closure $\overline{\phi(X)}$?

From T.A.Springer, Linear Algebraic Groups, the end of Chapter 1. Assume $X\rightarrow Y$ is a morphism of varieties. Using a covering of $Y$ by affine open sets, we reduce the proof to the case that $Y$ is affine. Similarly we can also assume $X$…
Bombyx mori
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Application of Hirzebruch Riemann Roch

Let $X,Y$ be smooth projective varieties over $\mathbb C$ where $X$ is the universal cover of $Y$. Assume that the fundamental group of $Y$ is finite and has order $d$. Then we want to show that $\chi (X)=d \chi(Y)$ where $\chi(\cdot)$ means Euler…
Bernard
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Noether normalization theorem

I'm reading: Hulek, "Elementary Algebraic Geometry", i can't understand a comment he does about Noether normalization theorem, which tells: Le $k$ be a field with infinitely many elements, let $A=k[a_1,\ldots,a_n]$ be a finitely generated…
Federica Maggioni
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Do regular functions on an algebraic variety separate points?

Define a variety as a reduced separated scheme of finite type over an algebraically closed field. Given two closed points $P$ and $Q$ on a variety, does there exists an open set containing them and a regular function on it, such that one of $f(P)$…
user665452
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A rational curve in the exceptional locus?

Let $f:X\rightarrow Y$ be a blow-up of a complex manifold $Y$ along a smooth submanifold. Is it easy to see that the exceptional locus contains a rational curve? Could anyone give a proof or suggest a reference for this fact?
Liu
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Dualizing Sheaf of $\tilde{\mathbb{P}}^{3}$

Let $\pi :\widetilde{\mathbb{P}^{3}} \longrightarrow \mathbb{P}^{3}$ be the blowup of $\mathbb{P}^{3}$ along a regular curve $\mathcal{C}$, with exceptional divisor $E$. We know the following: If $X$ is a projective nonsingular variety over an…
Allan Ramos
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How is it geometrically justified that the tangent space to $y(y-x^2)=0$ is the whole plane?

I'm reading Shafarevich's Basic Algebraic Geometry 1. For a variety $X\subset\mathbb A_k^n$ in affine space that goes through $0$, suppose its ideal is $I(X)=\langle f_1,...f_m\rangle$, and let $L=\{ta\mid t\in k, 0\not= a\in\mathbb A_k^n\}$ be a…
George
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question of Coherent sheaf (Hartshhorne book Example II.5.2.5)

Let $X$ be an integral Noetherian scheme, and $\mathcal{K}$ be the constant sheaf with the group $K$ equal to the function field of $X$ where the function field of $X$ is the residue field of generic point. Then, $\mathcal{K}$ is not coherent unless…
Sang Cheol Lee
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Proving That $ \text{Spec}\Big( k[x,y,t]/(ty-x^{2}) \otimes_{k[t]} k (a) \Big) \cong \text{Spec}\Big( k[x,y]/(ay-x^{2}) \Big) $

$ k $ is an algebraically closed field, and $ a \in k. $ This question stems from Example 3.3.1 in Hartshorne's Algebraic Geometry. There is a surjective morphism $ f: \text{Spec}\Big(k[x,y,t]/(ty-x^{2}) \Big) \longrightarrow \text{Spec}(k[t]), $…
Confused Student
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