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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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Kernel of the diagonal homorphism $f:B\otimes_A B\rightarrow B$

I'm studying Hartshorne's Algebraic Geometry book, and in the remark 8.9.2 I understood everything, besides one detail that is bothering me. He takes $U=SpecA\subset Y$, and $V=Spec B\subset X$, where $X$ and $Y$ are schemes, and a map…
ett
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meaning of normalization

I have seen the following construction and I would be very happy if someone could explain its meaning to me. We start from a smooth projective algebraic variety $Z$ over the complex numbers and a reduced effective divisors with simple normal…
norm
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FOAG exercise 7.4.D.

suppose $\pi:X\rightarrow \text{Spec}\ k$ is a quasifinite morphism. Show that $\pi$ is finite. (hint: deal first with the case where $X=\text{Spec}\ A,$ $A$ finitely generated over $k$. If $A$ contains an element $x$ not algebraic over $k$, we…
XiaYu
  • 587
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Self intersection of a curve?

I was reading wikipedia page on intersection theory, and it says that one can define self-intersection of a curve, and sometimes the number is negative. In the case there is a small deformation of the curve, say C' is a small deformation of C, one…
minimax
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Every $\text{Spec} R \rightarrow X$ factors through a open affine subscheme

For what kind of commutative rings $R$, the following property holds? For any scheme $X$ and any morphism $f:\text{Spec} R \rightarrow X$, there exists a open affine subscheme $U \hookrightarrow X$ such that $f$ factors through $U$. For example…
user395911
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Frobenius morphism not isomorphism of varieties

Let $k$ be a field of characteristic $p>0$ and $\phi:\mathbb{A}^1\to\mathbb{A}^1$ be the Frobenius morphism $\phi(x)=x^p$. According to Hartshorne exercise I.3.2, $\phi$ is not an isomorphism of varieties. Why is this? I think $\phi$ is the…
rpf
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local systems and Lefschetz pencils

Let $X$ be a smooth, projective algebraic variety over a field of characteristic zero. Let $U \subset X$ be an open subvariety such that $D=X \setminus U$ is a normal crossing divisor. Let $\mathcal{E}$ be a local system on $U$. I would like to know…
loclefs
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The "degree" of the Fano scheme of a projective variety vs. the degree of its "image" in projective space.

Let $X\subseteq\mathbb{P}^n$ be a projective variety. The Fano scheme $F_1(X)$ of lines $L\in\mathbb{P}^n$ contained in $X$ is a subscheme of the Grassmannian $\mathbb{G}(1,n)$. We can find the class $[F_1(X)]$ in the Chow ring of the Grassmannian.…
Rob Silversmith
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Irreducible fibers of a closed subset implies irreducibility

Let $f:X \to Y$ be a morphism of varieties and let $Z \subset X$ be a closed subset. Assume that $f^{-1}(p) \cap Z$ is irreducible and of the same dimension for all $p \in Y.$ Show that $Z$ is irreducible. Here is what I tried: Assume the contrary…
Ehsan M. Kermani
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Stable reduction, bad primes, bad reduction

What do the following terms mean in the context of elliptic (and Tate) curves? Stable reduction Bad primes, bad reduction
user48900
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How to compute intersection multiplicity?

My book is Hulek's Elementary algebraic geometry. He defines the intersection multiplicity of $C,C'$ (given by $f,g$ respectively) at $P \in \mathbb{P}_k^2$ by $I_P(C,C')=\dim_k \mathcal{O}_{\mathbb{P}_k^2,P}/(f,g)$ He gives an example: $C$ is given…
Gobi
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Sheaf of germs of regular functions on an affine variety

Let $k$ be an algebraically closed field. Let $X$ be a Zariski closed subset of $k^n$. Let $I(X) = \{f \in k[x_1,\dots,x_n]| f(p) = 0$ for every $p \in X\}$. Let $A = k[x_1,\dots,x_n]/I(X)$. Let $U$ be an open subset of $X$. Let $f\colon U…
Makoto Kato
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quotients of smooth projective varieties by finite groups

I know there is a plethora of literature on how to construct quotients by groups, but my situation is quite particular, so I would appreciate if you could give me some hints or bibliographical references. I'm interested in the following question:…
loren
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What are stalks of Spec $Z/(60)$?

What are stalks of (structure sheaf of) affine scheme Spec $\mathbb{Z}/(60)=\{(2),(3),(5)\}$? What are its global sections?
Tom
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Intersection numbers and multiplicities in Fulton's book

I'm reading Algebraic Curves by Fulton where the concept of intersection numbers is introduced on page 36. They give both a definition $$I(P,F\cap G) = \mathcal{O}_P(\mathbb{A^2})/(F,G)$$ And a characterisation using 7 properties. The 5th property…
Jens Renders
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