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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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Vector Space as a variety/scheme

http://www.math.ubc.ca/~reichst/edtotal.pdf In page 9 of this paper there is a notation that I don't understand: $W_d(k)$. Let $V$ be a variety, and $k[V]$ be its coordinate ring. Let $W_d$ be the sub (vector) space of $k[V]$ of polynomials up to…
take008
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Hartshorne's exercice V.1.2: $\deg C=C.H$

Let $H$ be a very ample divisor on a surface $X$ and let $C$ be any curve in $X$. The divisor $H$ gives an embedding $X\to \mathbb{P}^n$ which gives sense in the degree of the curve $C$: it is the highest coefficient of the Hilbert polynomial (ie…
Gabriel Soranzo
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Definition of Zariski Topology

Could someone explain to me, what is Zariski Topology? Under what condition a topology can be called Zariski Topology? Between the set $$V(E)=\{P \in \mathrm{Spec}(R)|E \subseteq P\}$$ and $$D(r)=\{P \in \mathrm{Spec}(R) | r \notin P \},$$ which one…
Blackoffe
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find birational maps of a hypersurface onto $\mathbb{A}^2$

Consider $$Y = V(y^2-x^3) \subseteq \mathbb{A}^2$$ Now, $\phi: \mathbb{A} \to Y, t \mapsto (t^2,t^3)$ is a birational map, but the pullback $\phi^\ast: K[x,y]/(y^2-x^3) \to K[t], x \mapsto t^2, y \mapsto t^3$, is not an isomorphism of…
user7475
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Reducibility of a singular hyperelliptic curve

Leq $\mathbb{F}_q$ be a finite field of characteristic $2$. Let $g(x)$ and $f(x)\in\mathbb{F}_q[x]$ be such that $g(x)$ is irreducible and ${g^{\prime}(x)}^2 f(x) \equiv {f^{\prime}(x)}^2 \mod{g(x)}$. Consider the polynomial $F(x,y)=y^2 + g(x) y +…
Ivan Pogildiakov
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Every non-generic point in a curve is closed?

Let $X$ be a scheme integral scheme of dimension 1. If $X$ is affine, then it is clear that every non-generic point is closed. I wonder if this is true in general. If not, is it true if we suppose that $X$ is a curve? (A variety of dimension 1.)
Gabriel
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Smoothness of morphisms gives an exact cotangent sequence

This is Vakil 21.2 S, self-study. We are to show that if $\pi: X \to Y$ and $\rho: Y \to Z$ are smooth morphisms of schemes, then the relative cotangent sequence $$\pi^*\Omega_{Y/Z} \to \Omega_{X/Z} \to \Omega_{X/Y} \to 0$$ is also left-exact. By…
Johnny Apple
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Ideal of finite intersection of algebraic sets

In general if $X_1$ and $X_2$ are two algebraic sets on $k^n$ with $k$ a field of characteristic zero, we have that $I( X_1 \cap X_2 ) = \sqrt{ I(X_1) + I(X_2) }.$ Is posible in general compute $I(X_1 \cap X_2 \cap \dots \cap X_n)$ in terms of…
user73577
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Vector bundles of rank $q+k$ in $K(Y)$, where $Y$ is a projective scheme of dimension $q$ over some field.

Given a vector bundle $G$ of rank $q+k$ on a projective scheme $Y$ of dim $q$ over some field,show that there exists an injection $0\rightarrow O_Y^k\rightarrow G(n)$ for some sufficiently large $n$. I met this question when considering $[G(n)]$…
XiaYu
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Whats the relation between the Veronese embedding and the isomorphism $\operatorname{Proj}A\cong\operatorname{Proj}A^{(d)}$?

In the book Introduction to Schemes, G. Ellingsrud and J. Ottem show (in section 6.7) that if $A$ is a graded ring and $$A^{(d)}:=\bigoplus_{n\geq 0} A_{nd}$$ then the natural inclusion $A^{(d)}\to A$ induces an isomorphism…
Gabriel
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A question regarding etale morphisms of affine varieties

Suppose that $p:Y\rightarrow X$ is a surjective etale morphism of affine varieties (both over an algebraically closed field of characteristic zero), and let $x \in X$. Then is there a subvariety $V$ of $Y$ containing $y \in p^{-1}(x)$, such that…
Bob G
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Why does the diagonal morphism of a map between affine schemes correspond to the following morphism of rings?

Suppose I have a morphism of affine schemes $(f, f^\sharp) : \operatorname{Spec} A \to \operatorname{Spec} B$, then my question is - why does the diagonal morphism $(\Delta_f, \Delta_f^\sharp) : \operatorname{Spec} A \to \operatorname{Spec} A…
Perturbative
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If $A$ is a ring, then why is $\operatorname{Proj} A[t] \cong \operatorname{Spec} A$?

The following is an example I saw in a book on Algebraic Geometry. Example: Let $A$ be a ring and consider $A[t]$, with the grading given by $\deg t = 1$ and $\deg a = 0$ for $a \in A$. Then the structure map gives an isomorphism…
Perturbative
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Two ramified covers $\Rightarrow$ reducible ramification divisor?

Let $X,X',X''$ be algebraic varieties and let $X''\to X'$ and $X'\to X$ be two ramified covers. Is the ramification divisor of the composition $X''\to X'\to X$ reducible?
Bonanza
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Characterization of line bundles in $D^b(Coh(X))$

For a complex algebraic variety $X$ (reduced and of finite type), consider $D^b(Coh(X))$, the bounded derived category of coherent sheaves on $X$. Question: (1) Is it true that $F\in D^b(Coh(X))$ is a line bundle (in degree $0$) if and only if…
Amanda Taylor
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