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Mathematics Stack Exchange

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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For any ideal $ \mathfrak{a}\subseteq A$, $I(Z(\mathfrak{a})) = \sqrt{\mathfrak{a}}$, the radical of $\mathfrak{a}$

I'm trying to prove that $I(Z(\mathfrak{a})) = \sqrt{\mathfrak{a}}$, where $\mathfrak{a}$ is an ideal of $A = K[x_1, ... , x_n]$ and $K$ is an algebraically closed field. In case this notation is nonstandard: if $T \subseteq A$ then $Z(T) = \{P \in…
Matt
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Question about dimension of varieties: where is this hypothesis necessary?

I found the following result on the web: Theorem. Let $f:X\rightarrow Y$ be a morphism of varieties, and assume that the dimension of all fibers $n=f^{-1}(P)$ is the same for all $P\in Y$. Then $\dim X=\dim Y+n$. Question 1: Should I also ask that…
Renan Mezabarba
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3
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Iitaka fibration over canonical model

I am looking for a referrence for the proof of following fact If the minimal projective manifold has positive Kodaira dimension and it is not of general type, it admits an Iitaka fibration over its canonical model
user61135
3
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1 answer

surjection between coherent sheaves on affine curve

I do not see the following statement: Suppose $\mathcal{F}$ is a locally free coherent sheaf on a smooth affine curve $X=\text{Spec }A$. Then for any coherent sheaf $\mathcal{G}$, there is an exact sequence $$…
Carsten
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2 answers

Upper semicontinuity of fibre dimension on the target

This is Vakil 18.1.C. Suppose $\pi : X \to Y$ is a projective morphism where $Y$ is locally Noetherian (or more generally $\mathcal{O}_Y$ is coherent over itself). Show that $\{y \in Y : \dim \pi^{-1}(y) > k\}$ is a Zariski-closed subset of $Y$. He…
SAK
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3
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Smooth Curves of genus 3

Let $X$ be a smooth projective curve of genus 3 over an algebraically closed field of characteristic 0. How do I show that any curve like this is hyperelliptic or a plane curve of degree 4? Why is $K(X)$ isomorphic to $k(t)[y]$ for $y^{2}=f$ with $f…
bundle125
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compute euler characteristic for pull back line bundle of finite map

i want to know if there is any formula to compute the euler characteristic for pull back line bundle of finite morphism, and the same question for blow up one point? Espesically for algebraic surface.
user42804
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When Is A Quot Scheme Reduced?

For my research, I would like to know whether a certain Quot scheme is reduced. Reading the thread How To Tell Whether A Scheme Is Reduced From Its Functor, I was disappointed to find that there's no easy way to read this information off the functor…
Cass
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Application of the Riemann-Roch theorem

If C is a quadric hyperelliptic curve ($g(C)=3$ and the canonical line bundle is very ample) contained in the two dimensional complex projective space and $K_C$ is the canonical line bundle of $\mathbb{P}^2$ restrected to my curve. I take four…
dario
  • 729
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Elementary algebraic geometry

Let $p(z,w)=z^2+w^2-zw+1,$and $Z(p)=\{(z,w)\in\mathbb{C}\times\mathbb{C}|\,p(z,w)=0\}.$ Is this variety irreducible? Is $Z(p)$ a connected subset of $\mathbb{C}\times\mathbb{C}$ ? (in usual topology of $\mathbb{C}\times\mathbb{C}$. Im not interested…
BigM
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first chern class of cotangent bundle

If $X$ is a smooth variety, how do we see that $c_1(T_X^*)$ is the canonical divisor? I can see this for the projective space: if $X=\mathbb{P}^n$, then the Euler exact sequence $$ 0\to O\to O(1)^{n+1}\to T_X\to 0 $$ gives $c_1(T_X)=(n+1)[H]$, and…
adrido
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Is every $\mathcal{O}_X$-module homomorphism $\mathcal{O}_X^{\oplus n} \to \mathcal{O}_X^{\oplus m}$ given by a matrix?

For $A$ a ring, we know any linear map from $A^{\oplus n} \to A^{\oplus m}$ is given by a $m \times n$ matrix. For $(X,\mathcal{O}_X)$ a locally ringed space (or scheme), is every $\mathcal{O}_X$-module homomorphism $\mathcal{O}_X^{\oplus n} \to…
Charles
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A question from Gathmann's notes on Algebraic Geometry.

This is question from Gathmann's notes on Algebraic Geometry. Let $$C_n=\{(x,y)\in\Bbb{C}^2;y^2=(x-1)(x-2)\dots(x-2n)\}\subset\Bbb{C}^2$$ Gathmann says that if we go in a circle around any of the points $1,2,3\dots,n$, "we go from one copy of the…
user67803
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insight into the definition of intersection multiplicity for two plane curves

Let $X,Y$ be curves of $\mathbb{A}^2$ given by irreducible polynomials $f,g$ respectively, where the ground field $k$ is algebraically closed. Then it is known that the dimension of $k[x,y]/(f,g)$ as a $k$-vector space is equal to the cardinality of…
Manos
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sections of birational proper morphism over an etale cover

Let $f: Y \to X$ be a birational proper morphism. Assume that every point of $X$ has an etale neighbourhood over which $f$ has a section. Is it true that $f$ is an isomorphism?
Adam
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