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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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Equivalent definition of rational map

On The Arithmetic of Elliptic Curves, Joseph H. Silverman, the definition of rational map is given by: Let $V_1$ and $V_2 \subseteq \mathbb{P}^n$ be projective varieties. A rational map from $V_1$ to $V_2$ is a map of the form $$\varphi : V_1…
6666
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Is a proper map of varieties $f:X\to Y$ an isomorphism if $f_Z: X\times_Y Z\to Z$ is an isomorphism for any closed, one-point subscheme $Z\subset Y$?

Let $f:X\rightarrow Y$ be a proper morphism of schemes of finite type over algebraically closed field $k$ (not necessarily of characteristic 0). Is is it true that $f$ is an isomorphism if $f_Z:X\times_Y Z\rightarrow Z$ is an isomorphism for every…
Intoks Liobein
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Fibers equal implies schemes equal in a neighborhood

Let $f:X \rightarrow Y $ be a morphism of locally Noetherian schemes. Let $Z$ be a closed subscheme of $X$ and suppose that there exists a point $y \in Y$ such that $Z_y=X_y$ as schemes. Show that if $Z$ is flat over $Y$ at $z \in X_y$ , then $Z$ is…
Heidar Svan
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A question on generic point and a question on Hartshorne

On page 134, Weil divisors, example 6.5.2, he said: "The divisor of $y$ is $2Y$, because $y=0$ implies $z^2=0$, and $z$ generate the maximal ideal of the local ring at the generic point of $Y$." I was stupid and can not figure this out. Can someone…
Yoshinobu Osawa
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Betti Numbers and the Néron-Severi group.

In the Paper "On the Mordell-Weil lattices" it is proved that the rank $\rho$ of the Néron-Severi group of a rational elliptic surface equals 10. Without any further explanation, it is stated that "for a rational surface, $\rho = b_2$ holds". I…
M. E.
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2 questions about the morphism $\mathbb{A}_k^{n+1}\backslash\{0\} \rightarrow \mathbb{P}_k^n$

An exercise in Ravi Vakil's algebraic geometry notes says "Make sense of the following sentence: $$\pi: \mathbb{A}_k^{n+1}\backslash\{0\} \rightarrow \mathbb{P}_k^n$$ given by $$(x_0,x_1...,x_n)\mapsto [x_0,x_1...,x_n]$$ is a morphism of schemes.'" …
bart
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Determining whether $\mathbb{C}[X,Y]/(P)$ is a domain, principal, or factorial from geometry.

I saw recently that $\mathbb{C}[X,Y]/(Y-X^2)$ is a PID because we can show it is isomorphic to the polynomial ring $\mathbb{C}[X]$. On the other hand, $\mathbb{C}[X,Y]/(XY)$ is not even an integral domain. I am wondering if we can say when…
JessicaB
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What does it mean geometrically for a graded ring to be generated in degree $1$?

This is a rather basic question that I have never really thought about until now, when I was forced to think about a sheafy version. This condition is needed for example to define the relative Proj construction, and I don't understand what exactly…
A. Thomas Yerger
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Coordinate free map to projective space

Given a line bundle $\mathcal{L}=\mathcal{O}_X(D)$ on a smooth complete curve $X$ (over $\mathbb{C}$), it's quite well-known that if the space of global sections has $k+1$ base point free, linearly independent sections $s_0, \cdots ,s_k$ (assuming…
Ehsan M. Kermani
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Jacobians of reducible curves

Let $C$ be a connected, reduced but reducible projective curve over an algebraically closed field. Denote by $\pi:\tilde{C}\to C$ its normalisation. There are two standard references for computing $J(C)=\rm{Pic}^0(C)$, namely Harris-Morrison's…
bey
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Is the closed immersion of a smooth subscheme of a smooth scheme, a smooth morphism?

Let $X$ be a smooth scheme over a (perfect if you want) field $k$. Let $Y$ be a closed subscheme of $X$ that is also smooth over $k$. Is the canonical closed immersion $i:Y \to X$ a smooth morphism?
GMRA
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Zariski dense contains Zariski open?

Let $U\subseteq\mathbb A_K^n$ be a Zariski dense subset (here, $K$ is an algebraically closed field). I was told that $U$ then contains a Zariski open and dense subset. However, I can't seem to come up with an easy proof. Edit: The above is wrong,…
user38451
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Kunneth formula

I am trying to understand the proof of the Kunneth formula, as described by Ravi Vakil's notes in 18.2.8 here: http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf#page=475 I will follow Ravi's notation. Why is that the the tensor product…
hwong557
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What is a "closed subvariety"?

I'm studying algebraic geometry from the classical viewpoint in which the Zariski topology takes center stage and schemes have yet to be invented. I sometimes see the term "closed subvariety" thrown around, but I can't find a proper definition for…
goblin GONE
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Calculation of intersection of a divisor with itself

I am reading the following paper on the Cox ring of $\overline{M}_{0,6}$ by Ana-Maria Castravet: http://arxiv.org/abs/0705.0070 I am stuck on an intersection-theoretic question, which appears as formula (9.1) in the above paper. It says the…
minimax
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