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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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Why polynomials maps are continuous in the Zariski topology

I didn't understand this proof: What exactly is $f^{-1}(Z)$? and what are these $h_i$ in $Z$? This proof is easy, I need just a little bit of clarification in these points. Thanks a lot.
user42912
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Why are torsion sheaves like effective divisors?

I'm very new to algebraic geometry, but am trying to read some papers, and am confused by a few things. Let $X$ be a projective surface. My first question is notational: (0) If an author writes $$ch_2(E) < c_1^2(E)$$ should I understand this to…
AlgGeomQ
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Bertini Theorem (Hartshorne book Thm II.8.18)

Let $X$ be a nonsingular closed subvariety of $\mathbb{P}^n_k$, where $k$ algebraically closed field. Bertini Theorem say that there exists a hyperplane $H \subset \mathbb{P}^n_k$, not containing $X$ such that $H\cap X$ is nonsingular. In proof, we…
Sang Cheol Lee
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Hartshorne Example II 4.0.1

This example wanted to show if $k$ is a field, $X$ the affine line with a double point as in Ex 2.3.6, then X is not separated. It argued that $X$(product over $k$)$X$ is affine plane with double axes and four origins. It is not closed because all…
user93417
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1 answer

Proving that 4 specified sets are not algebraic

I have four sets that I've come up with, which I think fail to be algebraic. However, I don't know how to prove this. They are: The graph of $\mathbb{C} \rightarrow \mathbb{C} : z \mapsto e^z$ {$(z, w) \in \mathbb{A}_\mathbb{C}^2 \mid |z|^2 + |w|^2…
user93779
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Is a union of varieties singular in the intersection of its irreducible components?

Consider a separated, reduced scheme $X$ of finite type over some algebraically closed field $\Bbbk$. Let $X=X_1\cup\cdots X_r$  be its irreducible components, each of which is then a variety. Assume that $P\in X_i \cap X_j$  for $i\ne j$. I then…
Jesko Hüttenhain
  • 14,792
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Global sections of Serre's twisting sheaf

Let $I_1$ and $I_2$ be homogenous ideals in $A:=\mathbb{C}[X_0,\ldots,X_n]$. Assume that $I_1 \subset I_2$. Let $X=\mathrm{Proj} (A/I_1)$ and $Y=\mathrm{Proj}(A/I_2)$. Then, 1) Is it true that the natural morphism $\Gamma(X,\mathcal{O}_X(n)) \to…
Chen
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Questions about skyscraper sheaves.

Let $X$ be a topological space and $P \in X$ a point. Let $A$ be an abelian group. The skyscraper sheaf $i_P(A)$ on $X$ is defined as follows: $i_P(A)(U) = A$ if $p \in U$ and $0$ otherwise. We need to verify that the stalk of $i_P(A)$ is $A$ at…
LJR
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Question of Hartshorne proposition II6.6

Let $X$ be a scheme which is a noetherian integral separated. In hartshorne's book, $X \times_\mathbb{Z}\mathbb{A}_\mathbb{Z}$ is also a noetherian integral separated. I understand $X \times_\mathbb{Z}\mathbb{A}_\mathbb{Z}$ is a noetherian and…
Sang Cheol Lee
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Is the projection from a Zariski product of Zariski closed subsets an open map?

Let $X \subseteq \mathbb A^n$ and $Y \subseteq \mathbb A^m$ be Zariski closed. Then the (Zariski) product $X \times Y \subseteq \mathbb A^{n + m}$ is closed and there is a projection map $p\colon X \times Y \to X$ which is continuous in the Zariski…
Jim
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Does a birational map of an affine variety extend uniquely to a birational map of its projective closure?

Let $V \subset \mathbb{A}^n$ be an affine variety, and $\varphi: V \to V$ a birational map. Does there exist a unique birational map $\varphi' : V' \to V'$ of the projective closure $V' \subset \mathbb {P}^n$ of $V$ that agrees with $\varphi$ on $V…
Ricardo Buring
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Self-intersection of a divisor $D$ with complete linear system $|D|$ without fixed component

Let $D$ be a divisor on an algebraic complex smooth projective surface $S$. Assume that the complete linear system $|D|$ is not empty and has no fixed component. Is it true that $D^2 \geq 0$, where $D^2$ is the self-intersection of $D$?
Francesco Genovese
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Hartshorne Lemma (I 3.6)

Let $X$ be a variety, $Y \subset \mathbb{A}^n$ an affine variety and $\psi:X \rightarrow Y$ a map such that $x_i \circ \psi$ is a regular function. We want to show that $\psi$ is a morphism of varieties. Since regular functions form a ring, then for…
Manos
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Ramification divisor and Hurwitz formula of higher dimensional varieties

Assume $X,Y$ are smooth varieties, $f: X \to Y$ is a separated morphism. Then it is claimed that there is Hurwitz formula: $$K_X \sim f^*K_Y + R$$ with $R$ an effective diviosr. I try to prove this result following the curve case as in Hartshorne IV…
Li Yutong
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Inverse image presheaf

Let $f:X\rightarrow Y$ be a continuous map of topological spaces, and $\mathscr{G}$ a sheaf on $Y$. So far I failed to come up with a simple example where the presheaf $f^{-1}\mathscr{G}$ on $X$ obtained via the direct…
ashpool
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