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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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Ramification of covering maps of curves and the étale fundamental group

I'm interested in finite galois covers $\varphi: Y \rightarrow X$ between smooth proper curves over an algebraically closed field of characteristic zero, which are étale outside a prescribed finite set $D$ of points of $X$. More precisely, I'd like…
Nils Matthes
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Why geometrically irreducible quartic curves in projective plane corresponds to an open subset of projective space of dimension 14

Here's exercise 9.5G from Ravi Vakil's Foundation of algebraic geometry Recall that the quartic curves in $\mathbb{P}^2_k$ are parametrized by a $\mathbb{P}^{14}_k$. Show that the points of $\mathbb{P}^{14}_k$ corresponding to geometrically…
chan kifung
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Is every complex affine/projective variety isomorphic/birational to one defined by an ideal $I\subset \mathbb{Z}[x_1,\dots,x_n]$?

For a while now I've been thinking about this question, but I have no idea how to go about it: Is every complex affine/projective variety isomorphic/birational to one defined by an ideal $I\subset \mathbb{Z}[x_1,\dots,x_n]$? So I'm interested in…
user2520938
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$\widetilde{(M \otimes_A N)} \cong \tilde{M} \otimes_{\mathcal O_X} \tilde{N}$

Let $A$ be a ring, $X = \textrm{Spec } A$, and $M$ an $A$-module. For $U$ open, define $\tilde{M}(U)$ to be the abelian group consisting of all $m_{\mathfrak p} \in \prod\limits_{\mathfrak p } M_{\mathfrak p}$ such that locally, $m_{\mathfrak p} =…
D_S
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Fulton, Algebraic Curves, Exercise 1.37

I am stuck trying to prove this: Let $k$ be any field, $F\in k[X]$ a monic polynomial of degree $n>0$. Then the residues $\bar{1},\bar{X},...,\overline{X^{n-1}}$ form a basis for $k[X]/(F)$ over $k$. I thought I had done it correctly but my…
Purple
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Morphism of algebraic groups defined over $\overline{\mathbb{Q}}$

My question is about the general theory of algebraic groups, on which I'm still far from comfortable. I want to understand a sentence in Milne's "Shimura varieties" http://www.jmilne.org/math/xnotes/svi.pdf , bottom of page 60. We consider a…
Yoël
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Closed subschemes of $\mathbb{P}^n_R$ ($R$ noetherian).

This is a follow up to my previous question: Zero subscheme of a section: Making computations. I want to show that every closed subscheme of $\mathbb{P}^n_R$ when $R$ is Noetherian is of the form $V(s)$ (i.e. the vanishing scheme of a section of a…
Abellan
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induced morphism on cohomology groups

Let $f: X \rightarrow Y$ be a morphism of varieties, and $\mathscr F$ be a sheaf (mamybe of Abelian groups) over $Y$. Does $f$ induce a morphism $H^i(Y,\mathscr F) \rightarrow H^i(X,f^{-1}\mathscr F)$? If it does, then how is this morphism defined?…
IzumiEternal
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Hilbert function on finite set of projective space

Let $X$ be the set consisting of four point in $\mathbb P^2$ and $h_X$ be the Hilbert function. If the four points are not collinear, \begin{equation*} h_X(m) = \left\{ \begin{array}{ll} 3, & if~ m=1;\\ 4, & if ~m \geq 2. \end{array}…
Sang Cheol Lee
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Local ring of variety at a point

Let $X=V(xy) \subset \mathbb{C}^2$ be the affine algebraic set. (coordinate axes) How can it be defined the local ring of $X$ at a point $P=(0,b)$ and $Q=(0,0)$? My textbook only defines local ring for irreducible varieties. So I don't know the case…
Gobi
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Smooth varieties are local complete intersection

I read in Harris Book that any Smooth variety is local complete intersection but I don't know why. I wonder what can one say about singular points in curves that are local complete intersection. Remember that a point $P$ of a variety is said to be…
Santiago Pinzón
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Rational function field of product affine varieties

Let $X, Y$ be affine varieties, we know that the coordinate ring of the product variety $X\times Y$ satisfies $k[X\times Y]\cong k[X]\otimes_k k[Y]$. My question is is it true that for rational function field, we also have $k(X\times Y)\cong…
AG learner
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When does the first right derived functor of $\otimes O_Y$ is zero?

Let $X$ be a variety, $V$ is a hypersurface in $X$, and $Y$ is a closed subvariety in $X$ which intersects with $V$ transverally. Suppose the corresponding closed embeddings are:$i:V \to X,\quad j: Y \to X$ . Then we have the following exact…
Li Zhan
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Branch locus of a projection

I'm looking for a formula or quick trick for computing the branch locus of a hypersurface under projection. My specific case: Let $V\subset\mathbb{P}^3$ be a surface of degree $d$, $p\in\mathbb{P}^3$ (it's actually a node of $V$, if that's…
Charles Siegel
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Playing with plane curves

Let $\phi: \mathbb{R}^1 \longrightarrow \mathbb{R}^2$ be the map given by $t \mapsto (t^2,t^3)$. I'm trying to show that any polynomial $f \in \mathbb{R}[X,Y]$ vanishing on the image $C = \phi(\mathbb{R}^1)$ is divisible by $Y^2-X^3$. And what…
mary
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