Mathematics Stack Exchange
Mathematics Stack Exchange

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
8
votes
0 answers

"First" results in algebraic geometry where schemes are needed

I'm interested where in the study of algebraic geometry, one really needs the full theory of schemes for the first time? Are there any results about varieties where using Hartshorne Ch1 type of machinery gets very tedious, but where schemes make the…
JT1
  • 653
8
votes
1 answer

Reduced scheme associated to a scheme - HAG II 2.2.3b

I found some difficulties proving this exercise from hartshorne's book. Let us first reduce to the affine case. Let $X = \mbox{Spec } A$ and define $X_{\mbox{red}}$ as in the exercise. For people who do not have the book nearby, it is defined as the…
bbnkttp
  • 1,909
8
votes
2 answers

Problem I.4.7 in Hartshorne

Let $X,Y$ be varieties and suppose we have points $P \in X, Q \in Y$ such that the corresponding local rings are isomorphic, i.e. $\mathcal{O}_{Q,Y} \cong \mathcal{O}_{P,X}$. Then the problem is to show that there exist open sets $P \in U \subset X,…
Manos
  • 25,833
8
votes
1 answer

Cardinality of variety

I'm trying to show that the cardinality of any variety of positive dimension is $ |k |$ where $k $ is the field being considered. This is part of exercise I.4.8 in Hartshorne's Algebraic Geometry: Show that any variety of positive dimension over $k…
PeterM
  • 5,367
8
votes
1 answer

Quotient of an affine variety by a finite group

I have worked through the proof of the statement that a quotient of an affine variety X always exists in case the group G acting on X is finite (see "Algebraic Geometry, a First Course" by Harris, page 124), and now I'm trying to show that this…
Paul
  • 1,232
8
votes
1 answer

Exercise 5.18(d), chapter 2. Hartshorne

In this exercise, suppose we start with a locally free sheaf $ \mathscr E $ of rank n over a scheme $\mathbf Y $, then corresponding to that we can associate $\mathbf V(\mathscr E^ \lor) $ and when we come back we get its sheaf of sections which is…
Suhas
  • 497
8
votes
3 answers

Proposition II.$3.2$ in Hartshorne

In Proposition II. (3.2) of Hartshorne book "algebraic geometry". I can't understand proof of part. Proposition: A scheme $X$ is locally Noetherian iff for every open affine $U = \rm Spec A$, $A$ is a Noetherian ring. It suffices to prove that if…
Med
  • 1,598
8
votes
1 answer

moving part and fixed part of a linear system

What is the definition of the "moving part" and "fixed part" of a linear system$|L|$? I think the fixed part should be defined to be the greatest effective divisor $F$ such that $D-F\geq 0$ for every $D$ in the system, and the moving part is the…
user93417
8
votes
1 answer

Some question on exactness of sheaves

Let $X$ be projective surface and $C$ be an irreducible nonsingular curve on $X$. For any curve $D$, I know that $$0 \rightarrow \mathcal{I}_D \rightarrow \mathcal{O}_X \rightarrow \mathcal{O}_D \rightarrow 0$$ Tensoring with $\mathcal{O}_C$ I saw…
Sang Cheol Lee
  • 3,119
8
votes
0 answers

projective varieties and locally trivial fibrations

Suppose $X,Y$ are varieties, $Y$ is projective and $f: X \to Y$ is a locally trivial fibration with fibre $\mathbb{P}^1$. Then there exists an open covering $\{U_i\}_i$ of $Y$ such that $f^{-1}(U_i) \cong U_i \times \mathbb{P}^1$ for each $i$.…
Balerion_the_black
  • 2,060
8
votes
0 answers

Variety with non-isomorphic Chow ring and Grothendieck group

Let $X$ be a smooth projective variety. With rational coefficients, the Grothendieck-Riemann-Roch theorem implies that the Chern character induces a canonical isomorphism $\mathrm{ch} : \mathrm{K}_0(X) \otimes \mathbf{Q}…
user314
8
votes
1 answer

Dimension of fiber product of subvarieties.

Let $X,Y \subset \mathbb{A}^n_k$ be be subvarieties of pure dimension r,s respectively, K a field. How could I show that $X \times_k Y$ is of pure dimension $r+s$? I am self-studying so anything is welcome. I have tried Noether normalization but I…
RogozjinRob
  • 151
8
votes
2 answers

Proving that $\operatorname{codim}(X,Y)=\dim(\mathcal{O}_{X,\eta})$

I have a problem with the following exercise: Let $X$ be a noetherian scheme and $Y \subset X$ an irreducible closed subset. I have to prove that $$ \operatorname{codim}(X,Y)=\dim(\mathcal{O}_{X,\eta}), $$ where $\eta$ is the generic point of $Y$.…
FedorBromov
  • 81
8
votes
1 answer

Again: Ample and very ample line bundles

I am still on my quest to design a simple example of an ample, globally generated line bundle which is not very ample and also understand the global sections of its smallest, very ample tensor power. Let $\Bbbk$ be some algebraically closed field,…
Jesko Hüttenhain
  • 14,792
8
votes
1 answer

Exercise II 2.8 on Hartshorne's Algebraic Geometry

There are many people asking about exercises on this book. I've tried to check existing similar questions as many as I can, but I can't promise to have read everything. Sorry if this is duplicated. Let $X$ be a scheme. For any point $x \in X$, we…
sunkist
  • 1,133
1 2 3
…
99 100