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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.

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Dimension of the product of affine schemes

Let $X=\mbox{Spec}A$ and $Y=\mbox{Spec}B$ be two affine schemes such that $A$ and $B$ are domains and finitely generated as $\mathbb{Z}$-algebras, and contain $\mathbb{Z}$. Consider their product over $\mathbb{Z}$: $X\times_\mathbb{Z}…
rfauffar
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Rational map from affine cone to projective scheme

I've been working through Vakil's MATH 216 notes and have ran into a wall when he discusses the affine cone of a projective scheme in section 8.2.12. Namely, if S is a finitely generated graded ring, the affine cone of ProjS is defined to be SpecS.…
Garnet
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Showing that $\mathbb{P}_k^2$ is birational to $\mathbb{P}_k^1 \times \mathbb{P}_k^1$

I wanted to check that there was nothing (roughly) wrong with my reasoning in showing that $\mathbb{P}_k^2$ is birational to $\mathbb{P}_k^1 \times \mathbb{P}_k^1$. First of all, I know that for two irreducible quasi-projective varieities $X$ and…
Aaron
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Exercise 1.13 of II in Hartshorne algebraic geometry.

The problem is following. 1.13 Espace Etale of a Presheaf. Given a presheaf $\mathscr F$ on $X$, we define a topological space $Spe(\mathscr F)$, called the espace etale of $\mathscr F$, as follows. As a set, $Spe(\mathscr F) = \bigcup _{P \in X}…
Jeong
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A set is not semialgebraic

A subset $A$ of $\mathbb R^n$ is called semi-algebraic if it can be represented as a finite union of sets of the form \begin{equation*} \{x\in \mathbb R^n\; |\; p_i(x)=0, q_i(x)<0\; \mbox{for all }i=1, \ldots, m\}, \end{equation*} where $p_i$ and…
Richkent
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Line subbundles of vector bundles on smooth curves

Let $V$ be a vector bundle of rank at least 2 on a smooth (integral, projective) curve $C$. We know that a global section of $V$ is the same as a morphism $\mathcal{O}_C \to \mathcal{V}$ (letting $\mathcal{V}$ denote the sheaf of sections), and that…
Jake Levinson
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Morphism of Affine Algebraic Variety

Is it true that a morphism of affine algebraic varieties is continuous in Zariski topology? How should I proceed? thank you
Myshkin
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pullback of rational normal curve under Segre map

Let $\nu:P^1 \rightarrow P^2$ be the veronese map of degree $2$, i.e. $[Y_0 : Y_1] \mapsto [Y_0^2 : Y_0 Y_1 : Y_1^2]$ and let $\sigma: P^1 \times P^2 \rightarrow P^5$ be the Segre map. Consider the image $\mathcal{X}$ of $\sigma \circ \nu$, i.e.…
Manos
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Isomorphism of varieties via coordinate rings

There is a result which I think is true but it's not written anywhere. Since I just began studying algebraic geometry, it's hard to figure out this by myself. Let $K$ be a field, $X\subset\mathbb{A}^n_k$ and $Y\subset\mathbb{A}^m_k$ be affine…
Integral
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Hartshorne Exercise V.1.8 - On NumX

This exercise is intended to show that Num$X$ (divisors modulo numerical equivalence) on a smooth, projective, integral surface $X$ is a finitely generated free abelian group, without using some big theorem of Neron and Severi. We assume the ground…
Cass
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Projective morphism, generated by global sections basic question

I have a very dumb question. Let $X = \mathbb{P}^2_k = Proj(k[x,y,z])$ where $k$ is algebraically closed. We have an invertible sheaf $\mathcal{O}(2)$ on $X$. Its space of global sections contains the elements $x^2, y^2, z^2, xy, yz, xz$. It…
David
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Prove that for each $p \in Y$ the quotient field of $O_p$ is isomorphic to the field $K(Y)$.

I'm reading Algebraic Geometry of Hartshorne. I have a question about a proof. First of all I'll write the important definitions that Hartshorne uses. Let $Y\subset \mathbb{A}^n$ be a quasi-affine variety. A function $f:Y\to \mathbb{k}$ is said to…
Miguel
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For a $k$-morphism $X \to Y$ to be determined by $X(\overline{k}) \to Y(\overline{k})$, does $X$ really have to be _geometrically_ reduced?

Question: Does the following hold? Let $k$ be a field, $X$ a reduced scheme locally of finite type over $k$, $Y$ any $k$-scheme and $f,g\colon X \to Y$ two $k$-morphisms. Then $f=g$ if and only if there exists an algebraically closed field…
c_c_chaos
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How to prove that a linear system is *not* base point free?

Assume I have a line bundle $\mathcal{L}$ on a projective, nonsingular variety $X$ over, say, an algebraically closed field (in fact, you may assume $\mathbb{C}$). Given global sections $s_0,\ldots,s_k\in\mathcal{L}(X)$, how to prove that they do…
Jesko Hüttenhain
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Exercise I 5.4 Hartshorne

I have no idea how to begin this exercise from the Hartshorne: If $Y$, $Z$ are two varieties of $\mathbb A^2$ given by the equation $f=0$ and $g=0$, the intersection multiplicity at $P$ is the lengh of the $\mathcal O_p$-module $M = \mathcal…
user171326
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