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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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Image of polynomial map, what's its coordinate ring?

I'm just trying to get some basic facts straight, Given a polynomial map $$F : \mathbb A^n \to \mathbb A^r,x \mapsto (f_1(x), \ldots, f_r(x))$$ with $f_i \in k[x_1,\ldots,x_n]$, I know that the image $Y \subset \mathbb A^r$ of $F$ need not be an…
Dario
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Can arithmetic and geometric genus be arbitrary?

I'm reading about genuses from Liu's book "Algebraic Geometry and Arithmetic Curves". There he defines arithmetic genus of a projective curve and geometric genus of a smooth projective variety. As far as I understand, a projective curve is a…
selfstudying
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How to decompose into irreducible components

I only know how to find the irreducible components when I know what the image is, but there are lots of equations that are hard to figure out their images, is there any systematic way to find the irreducible components?
Damon
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When will the support of a non-effective Cartier divisor be pure of codimension 1?

Let $X$ be a scheme and $D \in Div(X)$ a non-effective Cartier divisor. I am curious as to when $\text{Supp } D$ is pure of codimension 1, i.e all irreducible components are of codimension 1. So, three concrete questions that I have been having are…
user101036
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Some questions about Hartshorne chapter 2 proposition 2.6

In Hartshorne chapter 2 proposition 2.6, Hartshorne shows that there is a fully faithful functor $t:\mathcal{Var}\rightarrow \mathcal{Sch}(k)$ from the category of varieties over $k$ to the category of schemes over $k$. He proceeds as…
Wei Xia
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$\mathbb{P}(O\oplus O(-1))\simeq \mathbb{P}^1\times \mathbb{P}^1$?

Let $X=\mathbb{P}^1$. I am looking at $\mathbb{P}(O_X\oplus O_X(-1))$ and can see that it is the blow up of the projective plane at one point. I also see that it is a $\mathbb{P}^1$-bundle over $X$, but can't quite see if it's isomorphic to…
adrido
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Extension of Vector bundles is a Vector bundle?

I guess this is quite easy, but I don't see a counterexample: let $X$ be a noetherian scheme (maybe with more hypotheses, but I don't think this will change much), then I have the feeling that it is not true that every extension of a vector bundle…
Cyril
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Problem I.5.4(c) in Hartshorne

The problem asks to show that if $Y$ is a projective curve in $\mathbb{P}^2$ of degree $d$ and $L$ is a line such that $Y \neq L$, then $\sum_{P \in L \cap Y} (L \cdot Y)_P = d$. The solution given here…
Manos
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Are noetherian hypotheses necessary for the theory of the etale fundamental group?

The etale fundamental group, as explained in SGA 1 Expose 5 and various other notes I've read, always makes the assumption that the scheme $S$ (for which one intends to construct a fundamental group), is locally noetherian. How necessary is this…
oxeimon
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Dimensions of global sections of a divisor and its pullback

I doubt the following claim, but it seems that the proof of Theorem 10.2 (page 301, and one can download the book from libgen.org) in the book "algebraic geometry: an introduction to birational geometry of algebraic varieties" uses it: Let $V,W$ be…
Li Yutong
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Algebraic varieties that are isomorphic after a base change

Let $k$ be a field, $\overline{k}$ its algebraic closure. Suppose $X$ is an algebraic variety over $\overline{k}$. This means that $X$ is a scheme with a finite covering by open affine varieties over $\overline{k}$. The definition i'm using for an…
Fernandes
  • 63
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Finitely many singular points of an irreducible polynomial

let $k$ be a field, and consider an irreducible polynomial $f∈k[x,y]$. Let $S(f)$ denote the singular points of $f$ (points that are simultaneously zero on $f$, the $x$-derivative of $f$, and the $y$-derivative of $f$.) If $k$ is algebraically…
Alex
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Rank of derivative polynomial map equals dimension image?

I've been told that given a polynomial map $f:X\to Y$ in characteristic zero, there exists an open dense subset $U$ of $X$ such that for all points $x$ in $U$, the rank of the derivative of $f$ in $x$ equals the dimension of the image of $f$. Could…
Jasmine
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Proof of rigidity lemma

I am trying to understand in full details the proof of rigidity lemma as proved here http://staff.science.uva.nl/~bmoonen/boek/DefBasEx.pdf [Lemma 1.11, Pag. 12]. The statement in this reference is Lemma Let $X$, $Y$ and $Z$ be algebraic varieties…
Pgatti
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pullback of canonical divisor

Let $Y$ be a smooth variety of dimension $n$. Then I can get (a representative for) the canonical divisor class $K_Y$ on $Y$ by taking any rational $n$-form $\omega$ on $Y$ and taking its divisor of zeroes and poles, so $K_Y\equiv div(\omega)$. Now…
ykm
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