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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.

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Chern Class of a tensor product of coherent sheaves

If $\mathscr{E}$ is a vector bundle of rank $k$ and $\mathcal{L}$ is a line bundle, Then: $$c_{k}(\mathscr{E} \otimes \mathcal{L}) = \displaystyle \sum_{i = 0}^{k}\binom{r-l}{k-l}c_{1}(\mathcal{l})^{k-l}c_{l}(\mathscr{E})$$ Is there a similar result…
Allan Ramos
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How to write down a non-degenerate cubic surface in P^4

I want to do something very concrete: write down a smooth scheme of given degree and dimension in projective n-space. A natural way to go about this is to try and write down a complete intersection, but not all degrees/dimensions can be gotten this…
Tony
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Prove that any conic in $\mathbb{P}^2$ is normal.

A variety $X$ is called normal if for any point $P$$\in$$X$ ,the local ring $\mathbb{O}_P$ is an integrally closed ring. Show that any conic in $\mathbb{P}^2$ is normal. Here is my attempt: Use the the fact that any conic in $\mathbb{P}^2$ is…
user45955
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How to see $\operatorname{Spec} k[x]$ for non necessarily algebraic closed field $k$?

I know that $\operatorname{Spec} \mathbb{C}[x]$ can be identified with the set $\mathbb{C}\cup *$, where $*$ is a generic point via the correspondence $$ \prod_{i}(x-a_i) \leftrightarrow \{a_i\}_i , \ \ \ \ (0)\leftrightarrow *. $$ This…
M. K.
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Are ideal sheaves locally free sheaves?

Let $X$ be a smooth scheme of finite type. I think ideal sheaf of codimension 1 subscheme of $X$ is locally free as it is locally defined by one equation. What about higher codimension cases?
M. K.
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Constant Presheaf not necessarily a sheaf. Proof?

Let $\mathcal{X}$ be a topological space and $\mathcal{F}$ the constant presheaf, that assigns to each open set $\mathcal{U}$ of $\mathcal{X}$ the set $A$. The restriction map is the identity $A \rightarrow A$. I want to show that this presheaf is…
Manos
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Degree of a Divisor = Self Intersection?

This has been bugging me for some time now. I know that in certain cases that the self-intersection of divisors and it's degree are the same. Like hyerplanes in projective space. I sometimes read that certain degrees are actually defined as the…
Mike
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the fundamental exact sequence associated to a closed space

Let $(X,\mathcal O_X)$ be an algebraic variety. If $Y\subseteq X$ is a closed subset, then we can equip $Y$ with a structure of algebraic variety $(Y,\mathcal O_Y)$. The function $i:Y\rightarrow X$ is the usual immersion, moreover if $i_*\mathcal…
Dubious
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Prove that the natural map $\alpha : \text{Hom}(X,\text{Spec} A) \rightarrow \text{Hom}(A,\Gamma(X,\mathcal{O}_X))$ is an isomorphism

This is question 2.4 in Hartshorne. Let $A$ be a ring and $(X,\mathcal{O}_X)$ a scheme. We have the associated map of sheaves $f^\#: \mathcal{O}_{\text{Spec } A} \rightarrow f_* \mathcal{O}_X$. Taking global sections we obtain a homomorphism $A…
Rioghasarig
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How is the hyperplane bundle cut out of $(\mathbb{C}^{n+1})^\ast \times \mathbb{P}^n$?

[Question has been updated with more context and perhaps a better explanation of my question.] Source: Smith et al., Invitation to Algebraic Geometry, Section 8.4 (pages 131 - 133). First, a brief set-up, whose purpose will become obvious in a…
Daniel McLaury
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Computing the degree of a finite morphism $\mathbb{P}^n\to \mathbb{P}^n$

Let $k$ be an algebraically closed field. Suppose that $f\colon \mathbb{P}^n(k)\to \mathbb{P}^n(k)$ is a morphism of the form $f = [f_0:\cdots: f_n]$ where the $f_i$ are homogeneous polynomials of degree $d$ with no nontrivial common zeros. In this…
froggie
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Relationship between very ample divisors and hyperplane sections

This question is based on a line in the proof of corollary IV.3.3 in Hartshorne's Algebraic Geometry. The first line of the proof goes: "if $D$ is an ample divisor (on a curve $X$), then some multiple is very ample, so $nD\sim H$, where $H$ is a…
Misja
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In algebraic geometry, how do you explicitly find the strict transform?

Let $X = Z(xy - zw)$ in $\mathbb{A}^4$ and $Y = Z(x, z)$. If $\pi: B \rightarrow \mathbb{A}^4$ is the blow up of $Y$, then how can I find the strict transform of $X$ and the exceptional divisor? I honestly have no idea how to find the strict…
mathdragon
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Construction of the global $\mathbf{Proj}$

I have many questions about the very abstract concept of global $\mathbf{Proj}$. I am following Hartshorne's book Algebraic Geometry, where this concept is on II.7, page 160. Let $(X, \mathcal{O}_{X})$ be a noetherian scheme and $\mathcal{S} =…
rla
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To show a morphism of affine k-varieties which is surjective on closed points is surjective

This is a exercise from Ravi Vakil's Foundations of Algebraic Geometry, Ex 7.4.E. Assume Chevalley's theorem. Show that a morphism of affine $k$-varieties $\pi:X \rightarrow Y$ is surjective iff it is surjective on closed points (i.e. if every…
Ben
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