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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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Hodge diamond of complete intersections

Suppose we have a smooth complete intersection of hypersurfaces with degrees $d_1,...,d_r$ in some $\mathbb{P}^N$. This should be a surface and in certain situations a surface of general type. What can one say about the Hodge diamond? Or what is its…
user109227
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Does closed immersion imply projective morphism?

Here $X\to Y$ is a projective morphism means: $X\to Y$ factors through a closed immersion $X\to \mathbb{P}_{Y}^{m}$, and then followed by the projection $\mathbb{P}^{m}_{Y}\to Y$. I have no idea how to find this $\mathbb{P}_{Y}^{m}$.
Li Zhan
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Computing divisors on nonsingular curve

This is problem 8.2 from Fulton's Algebraic Curves. Let $X = C = V(Y^2 Z - X(X-Z)(X- \lambda Z)) \in \mathbb{P^2}, \lambda \neq 0,1$. Compute div$(x)$ and div$(y)$, where $x = X/Z$, $y = Y/Z$.
Pablolpool
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Is the fiber product of two irreducible schemes an irreducible scheme ?

If $X$ is an integral scheme, is it true that $ X $ x Spec $Z[t] $ over Spec $Z $ an integral scheme ? I know that it is reduced, but I could not show that it is irreducible.
Suhas
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Euler characteristic in Zariski vs. classical Topology

Let $X$ be a smooth projective, complex variety. Denote by $\bar X$ its analytification, i.e. $X$ with the "classical" topology of a complex manifold. Now do these spaces have the same Euler characteristic $\chi(X)=\chi(\bar X)$? Why (not)? Edit: To…
Jesko Hüttenhain
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Flat families of schemes (Exercize of Vakil's notes)

I'm studying Vakil's notes "Fundation of algebraic geometry" (http://math.stanford.edu/~vakil/216blog/FOAGmar2313public.pdf) and I have a problem understanding exercize 24.4.O (page 638). In fact if we consider the coordinate ring $$ A=…
ArthurStuart
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Some question about map of varieties

X,Y are varieties, and $A(X),A(Y)$ are coordinate rimg ,respectively. If$f:X\rightarrow Y$ is a finite surjective map, that is $f^{-1}(p)$ is finite for all $p\in Y$, then $A(X)$ is finitely generated $A(Y)$- module??
Sang Cheol Lee
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On affine local properties

In learning the third section First Properties of Schemes of the second chapter of Hartshorne, I heard someone mentioned Nike's lemma (I don't know if I spell the name right): if a scheme $X$ has a property $P$ affine locally (i.e. there is an…
user14242
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Very ample sheaf on a blowup

Suppose one wants to prove that a $1$-dimensional integral proper scheme over an algebraically closed field is projective. This is a step in how Hartshorne has you prove that any $1$-dimensional proper scheme (over an algebraically closed field) is…
Matt
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A Veronese map is a morphism?

My question is really simple, I'm beginning to study Algebraic Geometry and I'm still struggling to get the basic concepts. I would like to know if a Veronese map $v_{n,d}:\mathbb P^n\to \mathbb P^N$ is a morphism. Thanks
user42912
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About vector bundles on algebraic varieties

If $X$ is an irreducible algebraic variety (over $\mathbb C$), an algebraic vector bundle of rank $r$ over $X$ is a couple $(E,\pi)$ where $E$ is an algebraic variety and $\pi: E\longrightarrow X$ is a surjective morphism, with the following…
Dubious
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Finding the Vanishing Set of an Algebraic Set

We've been given the set $X = \{(t^3,t^4,t^5) \in \mathbb{A}^3 \mid t \in \mathbb{A}^1\}$ (where the underlying field $\mathbb{K}$ is infinite), and have been asked to show that $X = \mathbb{V}(J)$ where $J = \langle xz -y^2, x^3 - yz, z^2 - x^2 y…
lokodiz
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Covering projective variety with open sets $U_i$ such that $\pi^{-1}(U_i) \cong U_i \times \Bbb{A}^1$: How to improve geometric intuition?

I am looking at exercise II 6.3 of Hartshorne. In the first part, he asks to show the following. If $V \subseteq \Bbb{P}^n$ is a projective variety (over some field $k$), let $X = C(V)$ denote its affine cone in $\Bbb{A}^{n+1}$, and let…
user38268
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$\mathcal{O}_X(D) \cong \mathcal{O}_X$: Map doesn't agree on intersection

Let $D$ be the Weil divisor $D = -2[(x)] + [(x-1)] + [(x-2)]$ on $\Bbb{A}^1_k = \operatorname{Spec} k[x]$. I want to show that $\mathcal{O}_X(D) \cong \mathcal{O}_X$. To do this, it is enough to specify isomorphisms $$\mathcal{O}_X(D)(D(f_i)) \to…
user38268
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Dominant finite morphism and finite algebraic extension

I don't know how to prove the following proposition. If two varieties $X$ and $Y$ are irreducible, a morphism $\phi: X \rightarrow Y$ is dominant and finite, then $K(X)$ is a finite algebraic extension of $\phi^*K(Y)$. Here, $\phi^*: K[Y]…
ShinyaSakai
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