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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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Projective bundle is projective?

Let $\mathbb{P}(E)$ be a projective bundle over some smooth projective variety $X$, defined over $\mathbb{C}$ for definiteness. Then this bundle is also a smooth projective variety. Smoothness is clear from the trivialization, and it is also clear…
Karsten
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Why the unitary group is not a complex algebraic variety?

The question comes from Exercise 1.1.2 of the book "An Invitation to Algebraic Geometry". By definition the unitary group U(n) is the group of all complex matrix that satisfies $U^*U=I$. I know that since conjugate is involved the definition…
Xipan Xiao
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Looking for a smooth curve that is not rational

I am preparing for an exam in (mostly classical) algebraic geometry, and I have some preparatory questions, among which: Can you write the equations of any nonsingular curve in any projective space which is not rational? A problem with this…
Daniel Robert-Nicoud
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Smooth subvariety at smooth points

Let $f: X \to Y$ be a finite, surjective morphism of smooth, quasi-projective varieties over a field $k$ of characteristic zero. Let $p \in X$. Is it true that I can find a subvariety $X' \subseteq X$ of codimension one such that $p$ is a smooth…
Hans
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Why do only fixed points contribute to the Euler characteristic?

Let $G$ be an algebraic group with zero Euler characteristic, acting on a variety $X$ (over $\mathbb C$). I read some time ago that then the Euler characteristic of $X$ can be computed as $$\chi(X)=\chi(X^G),$$ where $X^G$ is the locus of fixed…
Brenin
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Theorem 5.1. Chapter I in Hartshorne Book's

I find difficulty to understand the proof of this theorem : Theorem : Let $Y\subseteq\mathbb A^n$ be an affine variety. Let $Ρ\in Y$ be a point. Then $Y$ is nonsingular at $Ρ$ if and only if the local ring $\mathcal O_P$ is a regular local ring. Let…
Med
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Exercise 3.7 Hartshorne

Problem. Show that any two curves in $\mathbb{P}^2$ have a nonempty intersection. This seems to follow immediately from the Projective Dimension Theorem, but I was wondering if anyone could provide a more 'elementary' proof? Thanks
V-B
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Vanishing of global sections with very negative twists.

Let $k$ be an algebraically closed field and let $\mathcal{F}$ be a coherent sheaf on the projective space $\mathbb{P}^n_k$. I would like to know when is it true that $$ H^0(\mathbb{P}^n_k, \mathcal{F}(m)) = 0 \qquad \text{ for } m<<0 $$ For…
Daniele A
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Is it enough to check closed immersion at closed points?

Let $f \colon X\to Y$ be a morphisms of algebraic varieties, which is a closed immersion in the topological sense. We also know that $f_x\colon \mathcal O_{Y,f(x)}\to \mathcal O_{X,x}$ is surjective for every closed point $x \in X$. Can we conclude…
Gabriel Furstenheim
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Hartshorne Page 150, Theorem 7.1

Theorem 7.1 (a) says that- If $\phi$: $X \rightarrow \mathbb P_A^n $ is an $A$- morphism, then $\phi^*(\mathcal O(1)) $ is an invertible sheaf on $X$, which is generated by the global sections $s_i=\phi^*(x_i) $, $i$=0,1,...,n. I do not know how to…
Suhas
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Canonical and terminal singularities

Let $Y$ be a normal variety such that $K_Y$ is $\mathbb{Q}$-Cartier, and $f:X\to Y$ a resolution of singularities. Then, $$ K_X = f^\ast(K_Y) +\sum_i a_iE_i $$ where $a_i \in \mathbb{Q}$ and the $E_i$ are the irreducible exceptional divisors. Then…
Derek Allums
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Why is shefication necessary in constructing the reduced scheme?

Let $X$ be a scheme. Hartshorne defines the reduced scheme associated to $X$ as the sheafication of the presheaf $U \mapsto \mathcal{O}_X(U)_{\text{red}}$. Is there any example that shows that this presheaf need not be a sheaf?
user115940
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How does a hyperplane in the projective space corresponds to the Twisting Sheaf of Serre

Reference: Hartshorne,Chapter 2, Proposition 6.17 $X= \mathbb P ^n _k$ for some field k. Then the generator of the $Cl (X)$ (which is the group of weil divisors modulo principal divisor) is generated by a hyperplane which corresponds to the…
Babai
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Restriction of scalars is a right adjoint to tensor product

This is the first time I'm asking a question, hope it's not a silly question. I'm studying through Ravi Vakil's notes, and I came up to this 1.5E exercise that reads like this: Suppose $A \to B$ is a morphism of rings. If $M$ is a $B$-module, you…
Tyche
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Hartshorne proposition II(2.6)

I am studying the proof of the following proposition in Hartshorne - Let $k$ be an algebraically closed field. There is a natural fully faithful functor $Var(k)\longrightarrow Sch(k)$ from the category of varieties over $k$ to schemes over…
gradstudent
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