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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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The pre-image of a curve under a surjective morphism

The following lemma and proof come from the paper Toward a numerical theory of ampleness, Chapter I, Section 4, Lemma 1 by Kleiman: Let $f: V'\rightarrow V$ be a surjective morphism from a proper irreducible variety $V'$to a integral curve $V$.…
Hobo
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Base change for proper morphism in Hartshorne

After reading Section III.11 (The theorem on Formal functions) and III.12 (The semicontinuity Theorem), I feel that I get some kind of formalism instead of a clear picture of what is going on. So the big question is how one should understand these…
Jiangwei Xue
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Hartshorne, Chapter 1, Projective varieties, Question 4(b)

Chapter 1, section 2, question 4(b) in Algebraic Geometry says An algebraic set $Y \subseteq \mathbb{P}^n$ is irreducible iff $I(Y)$ is a prime ideal. I'm confused about the solution given in…
Kumquats
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Picard group of $G(k, n)$ saying about automorphisms

Let $G(k, n)$ be the Grassmannian of $k$-dimensional subspaces of $K^{n}$, $K$ a field, embedded in $\mathbb{P}^{N}$ by the Plücker embedding. In Harris' Algebraic Geometry, A First Course, Theorem 10.19 states that $$\mathrm{Aut}(G(k, n)) =…
rla
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Jacobian criterion for singularity types

Let $I = (f_1,...,f_m) \subset k[x_1,...,x_n]$ be a prime ideal where $k$ is an algebrically closed field and let $V = \text{Spec}(k[x_1,...,x_n]/I)$ be the corresponding algebraic variety. If we take any $p \in V$, we can define the Jacobian…
KSAKY
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Non effective divisor on a smooth quartic surface

Let $E$ be a nonzero, non effective divisor on a smooth quartic surface $X \subset \mathbb P^3$ (i.e. $H^0(X, E) =0$). Then is it possible that $H^1(X, E) \neq 0$? Euler Characteristic computation suggests that it is possible iff $E^2 \neq -8$. Are…
Proj
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Steps in proof of Proposition 2.4.17 in Liu

I am going through the proof of Proposition 4.17 in Liu's Algebraic Geometry and Arithmetic Curves: First question (SOLVED): I can't verify why $V(f\vert_U)$ contains $W\cap U$. If we write $\phi:W\cong\operatorname{Spec}A$ and…
Sha Vuklia
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Asymptotic Riemann-Roch formula with exceptional divisor involved

Let $X$ be a projective surface, let $D$ be a curve on $X$ viewed as a Cartier divisor. Let $P$ be a point on $D$, and let $\pi: \tilde{X} \to X$ be the blow up $X$ at $P$ with exceptional divisor $E$. We know that $\pi^*D.E=0$ and $E^2=-1$. For…
finiteness
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Morphism and slope of bundles on $\mathbb P^2$

Let $H$ be a very ample line bundle on $\mathbb P^2$ given by $\mathcal O_{\mathbb P^2}(1)$. Let $E$ be a rank $r$, slope stable (w.r.to $H$) vector bundle on $\mathbb P^2$. Is it true that there does not exist any nontrivial homomorphism between…
Sherlock
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Line bundle determined by a linear surjection $V \to V/W$

I am new to things like line bundles and have just been reading page 10 of this document here. Now I have some elementary questions: Given a finite dimensional complex vector space $V$ and a one - dimensional vector space $W$, we can form $V/W$…
user38268
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How to find a non-singular hypersurface in $\mathbb P^n$ containing a certain curve

Assume that $X$ is a non-singular projective curve in $P^n_k$, where $k=\bar{k}$ and $n \geq 3$. Prove that for every $m$ large enough, there always exists a non-singular hypersurface of deg $m$ containing $X$. The hardest part of this problem is…
user884626
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Hartshorne AG, proof of proposition 5.7 and 5.8: "The question is local, so we may assume $X$ is affine ..."

On several instances, Hartshorne states that certain questions are "local". What is the reasoning behind it? Some instances in the text by Hartshorne, algebraic geometry, occur on p. 114 f., proof of proposition 5.7, or proof of proposition 5.8. In…
user823
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$V(f)\subset V(f,g)$?

I didn't understand this part in this proof: For me, we have to have the contrary $V(f,g)\subset V(g)$. Maybe the author made a mistake. Thanks in advance.
user42912
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mapping of Zariski tangent spaces

Let $r: X\to Y$ be a morphism of algebraic sets (over $\mathbb C$) and let $X'$ be an algebraic subset of $X$ containing $x_0.$ It is easy to see that if $r_*: T_{x_0} X\to T_{r(x_0)} Y$ is a monomorphism then its restriction to $T_{x_0} X'\to…
Adam
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If $K$ is finite, then every subset of $\mathbb A^n(K)$ is algebraic

I'm trying to prove that if $K$ is a finite field, then every subset of $\mathbb A^n(K)$ is algebraic. I know that if $K$ is finite, then every element of $K$ is algebraic, i.e., for every $a\in K$ there is a polynomial $f\in K$ such that $f(a)=0$,…
user42912
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