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
4
votes
2 answers

If $X$ and $Y$ are finite etale covers of each other, are they isomorphic?

This is a somewhat strange question perhaps, but I was wondering the following: Suppose $X$ and $Y$ are schemes (lets say Noetherian, and integral as well if you'd like). Suppose that there are maps $f: Y \rightarrow X, g: X \rightarrow Y$ which…
WeatherMan
  • 499
4
votes
1 answer

Flat but not very flat families

I'm doing an exercise in Hartshorne's Algebraic Geometry, Ex 9.5 in Chapter III, whose part (a) states the following: Given an example to show that if $\{X_t\}$ is a flat family of closed subschemes of $\mathbb{P}^n$, then the projective cone…
hyyyyy
  • 350
4
votes
1 answer

Pushforward of constant sheaf on the generic point.

Let $X$ be an integral scheme, and $g: \eta \to X$ the inclusion of its generic point. If $M$ is an abelian group and $M_\eta$ is a constant sheaf on $\eta$ taking value $M$, then is it true that $g_* M_\eta = M_X$? My thought is yes, as if $U…
DKS
  • 1,280
4
votes
1 answer

Finite-dimensionality of $H^1(X,\mathscr O_X)$ for a projective curve

A well-known result by Serre is that properness of a noetherian scheme $(X,\mathscr O_X)$ over $k$ implies finite dimensionality of $H^i(X,\mathscr O_X)$ for all $i \geq 0$. For a projective variety it is easy to prove that $H^0(X,\mathscr O_X) =…
user512346
4
votes
1 answer

Is the set rational points on the unit circle isomorphic to $\mathbb{Q}$ as affine varieties?

I will denote by $X$ the set of rational points on the unit circle, i.e., $$X := \{ (x,y) \in \mathbb{Q}^2: x^2 + y^2 = 1 \}.$$ Viewing both $\mathbb{Q}$ and $X$ as an affine varieties, then every morphism $f: \mathbb{Q} \to X$ is given by two…
Yuhang Chen
  • 827
  • 4
  • 9
4
votes
1 answer

compute degree of a line bundle

I have a proper family of projective curves $X\rightarrow S$ with $S=Spec(k[t])$ where $k$ is a field and $t$ an indeterminate (can assume $k$ alg.closed if you like). Assume that the fiber $X_t$ over $t=0$ is a divisor with 3 smooth irreducible…
kekko
  • 53
4
votes
2 answers

Solutions of $\frac{1}{\cos \theta} = a \sin \theta - b$

One of my math professors and I are working on a physics problem involving spinning a chain, and we decided to go as simple as possible and work out the solution explicitly for that case (a long rod hanging from a hinge rotating in a horizontal…
El'endia Starman
  • 3,988
4
votes
1 answer

Divisor section correspondence

Let $X$ be a smooth variety over $\mathbb{C}$, and let $D$ be a divisor on $X$. What is the condition on $D$ so that we can speak of a canonical section $s$ on $H^0(X,D)$ such that $D$ is the zero locus of $s$? Thanks.
minimax
  • 1,013
4
votes
1 answer

Are all Projective Space Bundles on Schemes Projectivizations of Vector Bundles?

I'm taking a course on complex manifolds, and in class we saw this fact for complex manifolds. I have a moderate background in algebraic geometry, and was interested in the same question for $\mathbb{P}^n$ bundles over a scheme $X$, where now we…
A. Thomas Yerger
  • 17,862
  • 4
  • 42
  • 85
4
votes
1 answer

There is a bijection between irreducible components of the generic fiber and irreducible components passing through it.

I have been working on the following problem from Ulrich-Görtz today and I can't seem to find a nice solution. Let $f:X \rightarrow Y$ be a morphism of schemes and let $Y$ be irreducible and $\eta$ the generic point of $Y$. Show that there is a…
Dedalus
  • 3,940
4
votes
1 answer

Hartshorne Exercise II.3.17 Noetherian Induction

I'm confused by the Noetherian Induction exercise in Hartshorne. Let $X$ be a Noetherian topological space, and let $\mathscr{P}$ be a property of closed subsets of $X$. Assume that for any closed subset $Y$ of $X$, if $\mathscr{P}$ holds for…
nekodesu
  • 2,724
4
votes
2 answers

Surjective closed morphism of schemes induces inequality of dimensions

Let $f: Y\to X$ be a surjective closed morphism of schemes. Prove $\dim Y\geq \dim X$. I was made to believe that this should be easy, but I somehow did not manage to prove this. Using the fact that $f$ is closed and surjective, it is easy to…
asdq
  • 3,900
4
votes
1 answer

What is a "very bad" singularity?

In page 94, example 4.42 of Milne's notes on algebraic geometry, he mentions that the singularity at $(0,0)$ is "very bad". The surface is $ V: Z^3 = X^2Y $, which has singular locus $ X = Z = 0 $. What does he mean by this? The tangent space is $…
user564167
4
votes
1 answer

Hartshorne Exercise I.4.3: rational functions and regular functions.

Let $f$ be the rational function on $\mathbb{P}^2$ given by $f = x_1/x_0$. Find the set of points where $f$ is defiend and describe the corresponding regular function. My question: I know that the set of points where $f$ is defined is $\{[1: x_1:…
nekodesu
  • 2,724
4
votes
3 answers

Hartshorne Exercise 1.1.10: Give an example of a noetherian topological space of infinite dimensions

Hartshorne Exercise 1.1.10: Give an example of a noetherian topological space of infinite dimensions. I'm baffled by why such space can exist. Instincts told me that I shouldn't take the Spec of any Noetherian ring because then prime ideals have…
nekodesu
  • 2,724
1 2 3
…
99 100