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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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How to show that $\mathcal{O}(1)$ is generated by the global sections $x_1, \ldots, x_n$?

On page 150 of Algebraic Geometry by Hartshorne, line 4 of paragraph 2, it is said that $\mathcal{O}(1)$ is generated by the global sections $x_1, \ldots, x_n$. How to show that $\mathcal{O}(1)$ is generated by the global sections $x_1, \ldots,…
LJR
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Morphic Image of a Complete Variety

A variety $X$ is called separated if $\Delta_X = \{ (x,x) \mid x \in X \} \subset X \times X$ is closed in $X \times X$. A variety $X$ is called complete if it is separated and for any other variety $Y$ and a close subset $Z \subset X \times Y$ we…
LinAlgMan
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Hartshorne's t functor

Hartshorne (II Prop 2.6) defines a functor $t$ from the category of topological spaces to itself as follows: If $X$ is a topological space, define $t(X)=\{Z\subseteq X:Z\text{ is irreducible and closed}\}$. Assign $t(X)$ the topology with closed…
Avi Steiner
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Separated morphism - Hartshorne Corollary II 4.2

The corollary says - An arbitrary morphism $f:X\longrightarrow Y$ is separated if and only if the image of the diagonal morphism is a closed subset of $X\times_{Y} X$. I am studying the proof of this proposition. One way is obvious. To prove the…
gradstudent
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Proof of Proposition IV.3. 8 in Hartshorne

Hartshorne book Proposition (IV.3. 8) is that Let $X$ be a curve in $\mathbb{P}^3$, which is not contained in any plane. where, curve means a complete, nonsingular curve over algebraically closed field $k$. Suppose either (a) every secant of…
Sang Cheol Lee
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Prove that $H^1(\mathbb P^n,T_{\mathbb P^n})=0$

There is an exercise in Ravi Vakil's notes, namely exercise 21.5.Q, asking to prove that $H^1(\mathbb P^n,T_{\mathbb P^n})=0$, where $T_{\mathbb P^n}$ is the tangent bundle of the projective space. I would like a hint on how to do this. I started by…
Brenin
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Image of a diagonal morphism.

I was trying to study the definition of Sheaves of Differentials of Hartshorne p.175. It says the diagonal morphism $\Delta:X \rightarrow X \times _Y X $ gives an isomorphism of $X$ onto its image, which is locally closed sub-scheme of $X \times _Y…
Babai
  • 5,055
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Why degree of a reducible projective variety is the sum of the degree of its irreducible components

Could anyone show me how to prove that The degree of a reducible projective variety is the sum of the degree of its irreducible components? The definition of the degree I know is quite vague, saying that the degree of a projective variety $X$ is…
hxhxhx88
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Is an ample divisor linearly equivalent to a non-trivial effective divisor

In other words, if $L$ is an ample divisor, must we have $h^0(L)>0$ or can this be zero?
Heitor Fontana
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Proving that maximal ideals in coordinate ring determine unique point

I have the following homework question: Let $X \subseteq \mathbb{A}^n$ be an irreducible algebraic subset, and let $\mathbb{K}$ be algebraically closed. Show that every maximal ideal in $\mathbb{K}[X]$ determines a unique point $p \in \mathbb{A}^n$…
lokodiz
  • 2,340
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If $p_i:X\rightarrow Y_i$($i=1,2$) are immersions, is $X\rightarrow Y_1 \times Y_2$ immersion?

If $p_i:X\rightarrow Y_i$($i=1,2$) are immersion of $S$-schemes, is $X\rightarrow Y_1 \times_{S} Y_2$ an immersion? I don't know if it is true.I tried to treat the affine case, $X,Y_i,S$ are spectrum of rings$A,B_i,C$. But I don't know what the…
user93417
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Variant of Serre's criterion for affineness

Let $X$ be a scheme. You may assume that it is nice enough, perhaps of finite type over a field $k$ and smooth. In my application $X$ is not separated, a priori. Assume that every coherent sheaf on $X$ is generated by global sections. Does it follow…
Martin Brandenburg
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Euler character of etale finite cover

Let $\pi: \tilde{X} \to X$ be an etale finite cover, then why the Euler character has relation: $$\chi(\tilde{X},\mathcal{O}_{\tilde{X}})=\deg(\pi)\chi({X},\mathcal{O}_{{X}}).$$ I try to use Riemann-Roch, but do not know how to relate Chern…
Li Yutong
  • 4,065
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Ramification points and Weierstrass points

This is question 7.4.7 out of Liu's book: Let X be a hyperelliptic curve of genus $g \geq 2$ endowed with a separable morphism $f: X \rightarrow \mathbb{P}^1_k$ of degree 2. We can write $K(X) = k(t)[y]$ with a relation $y^2+Q(t)y=P(t)$ .Let $x_0…
Tedar
  • 529
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If sections of certain sheaves agree on an open neighbourhood does it agree globally?

Let $X$ be a non-reduced scheme whose associated reduced scheme is smooth. Assume as well that $X$ is irreducible. Suppose that two global sections of the tangent sheaf of $X$ agree on a non-empty open set in $X$. Does this mean that the two global…
Chen
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