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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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Section of direct-sum and tensor product of line bundles

Suppose I have a variety $X$ and suppose that I have three line bundles $L_1, L_2$ and $L_3$ over $X$ such that $L_1 \oplus L_2 \oplus (L_2 \otimes L_3)$ has a nonzero global section. Does it imply that we have a nonzero global section of the bundle…
jack
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Confusion on Ravi Vakil FOAG 15.2.E (b) (09.2022 version)

15.2.E. Suppose $X$ is a normal irreducible Noetherian scheme, and $\mathscr{L}$ is an invertible sheaf, and $s$ is an nonzero rational section of $\mathscr{L}$. (a) Describe an isomorphism $\mathscr{O} ( \operatorname{div}s) \leftrightarrow…
Goldmund
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Prove the complement of a point in $\mathbb{A}^n$ is compact.

I am learning algebraic geometry and I came across the folowing question. Prove that the complement of a point is compact in $\mathbb{A}^n$. Does anyone know how to do this?
dunkindonuts
  • 143
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"Actual" points vs rational points of a scheme over a field.

Let $X$ be a scheme over a field $k$, and consider a point $x\in X$, i.e. a point in the underlying topological space of $X$. Does there exist a field extension $K$ of $k$ such that $x\in X(K)$? I have a hard time relating the notions of "actual…
mathfan24
  • 560
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transcendental base extension

An exercise in Hartshorne claims that a scheme $X$ of finite type over a field $k$ is geometrically irreducible (respectively geometrically reduced) if and only if $X \times_k K$ is irreducible (respectively reduced) for any field extension $K/k$.…
Justin Campbell
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Maps between projective varieties given in coordinates by homogeneous polynomials of the same degree not simultaneously vanishing are morphisms

Let $\varphi: V \to W$ be a map between projective varieties $V \subset P^n$ and $W \subset \mathbb{P}^m$ given by $\varphi([x_0 : \ldots : x_n]) = [\varphi_0([x_0 : \ldots : x_n]): \ldots : \varphi_m([x_0 : \ldots : x_n])]$, where the $\varphi_i$…
Ricardo Buring
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Confusion on Vakil's universal property of pullback, Vakil 16.3.A

I'm having some trouble wrapping up the proof for 16.3.A in Vakil's FOAG, which basically asks to show that the tensor product construction of $\pi^{\ast} \mathscr{G}$ (for $\pi : X \to Y$, $\mathscr{G}$ quasicoherent on $Y$) satisfies the following…
cdsb
  • 418
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$\bigcap_i \operatorname{Ann} \tilde{(M_i)} = \operatorname{Ann} \tilde{M}$?

I’m reading Görtz–Wedhorn’s Algebraic Geometry, proof of the Proposition 7.34, and some question arises. Let $X$ be a scheme and $\mathcal{F}$ a quasi-coherent module of finite type. Define the annihilator $\newcommand{\Ann}{\operatorname{Ann}} \Ann…
Plantation
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parametrizing a circle near the origin

In the book "Using Algebraic Geometry" by Cox and co-workers, i encountered the following example: consider the circle with center $(-1,0)$ and radius $1$, given by the equation $x^2+2x+y^2 = 0$. Then the authors say "a parametrization of the circle…
Manos
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Double dual of a subsheaf of a vector bundle

Let $F$ be a non-trivial proper subsheaf of a vector bundle $E$ over a smooth projective surface $X$. Is it necessarily true that $F^{**} \subset E$? I can construct a map from $F^{**} \to E$ from dualizing twice the sequence $0 \to F \to E \to E/F…
Sherlock
  • 441
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Canonical sheaf of product of two smooth projective curves

Let $C_1, C_2$ be two smooth projective curves of genus atleast $2$. Let $X:=C_1 \times C_2$. If $P_1, P_2$ are denoted as the first and second projections, then it is known that the canonical bundle relation is given as follows : $K_X :=…
Proj
  • 95
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Is ideal sheaf left exact

Let $P$ be a zero dimensional scheme on a projective surface $X$. One can consider its ideal sheaf $I_P$ in $X$ to be the kernel of the morphism between $\mathcal O_X$ and pushforward of $\mathcal O_P$. Is the following true : the injection…
New
  • 347
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Dimension of Zariski Tangent over $\mathbb{Z}[2i]$

I'm trying to compute the dimension of the Zariski Tangent space of the affine scheme $\operatorname{Spec} A$, where $A = \mathbb{Z}[2i] \cong \mathbb{Z}[x]/(x^2 + 4)$, at the point $(2, 2i)$. Using the isomorphism in the previous sentence, we…
cdsb
  • 418
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Showing that a map $\operatorname{Div}(X) \to \bigoplus_{x\in C^{1}}\operatorname{Div}(\mathcal{O}_{X,x})$ is surjecitve (Gortz, Algebraic Geometry)

I'm reading the Gortz's Algebraic Geometry, proof of the Proposition 15.26 and stuck at understanding that some map appearing in the proposition is surjective. Let $X$ be an absolute curve ; i.e., a non-empty noetherian scheme with irreducible…
Plantation
  • 2,417
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Confusion on morphisms to projective spaces

Hartshorne's theorem II.7.1 claims that if $\phi : X\rightarrow \mathbf{P}^n_k$ is a morphism of a $k$-scheme into projective space, then $ \phi^*(\mathcal{O}(1)) $ is an invertible sheaf generated by global sections $s_i = \phi^*(x_i).$ Vakil's…
user960774
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