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
1 answer

What is $\operatorname{Pic}(\mathbb{P}^n_{\mathbb{Z}})$?

I would like to know the Picard group of the projective spaces over the integers $\mathbb{Z}$. I know that the projective space over a field $k$ has $\operatorname{Pic}(\mathbb{P}^n_{\mathbb{k}}) \cong \mathbb{Z}$, but what in the case of the…
user110071
  • 385
  • 1
  • 13
4
votes
3 answers

Are the local rings at smooth points of an irreducible variety isomorphic?

If it is wrong, then how about restrict the varieties to smooth ones? In order to investigate it I also asked whether Aut(X) always acts transitively over a projective variety X. But I don't know how to prove/disprove this either... This latter…
Honglu
  • 583
4
votes
0 answers

Fundamental group of family of rational varieties

Let $X$ be a smooth, projective, simply connected variety over a field $k$ (i.e. $\pi_1^{\text{et}} = 1$). Let $f: Y \to X$ be a family of rational varieties parametrized by $X$, such that $Y$ is smooth and projective. Then $Y$ should be simply…
Evariste
  • 2,620
4
votes
1 answer

Is a morphism with finite fibers birational?

Let $f: X \rightarrow Y$ be a morphism of projective varieties such that its fibers have finitely many points. Is $f$ birational on its image? Thanks.
rla
  • 2,345
4
votes
0 answers

Intuition for fiber of rational function sheaf

Let $X$ be integral scheme and $\mathcal K$ sheaf of rational functions on $X$. For any point $y\in X$ different of generic point we know that fiber of $\mathcal K$ (defined as usual as $\mathcal K _y / \mathcal m_y \mathcal K_y$) is zero. I'll be…
evgeniamerkulova
  • 2,062
4
votes
1 answer

Dimension of some moduli spaces

I expect that this is a very easy question, but somehow I can't get it. What is the dimension of the moduli space of complete intersections of degree 2 and 4 in $\mathbb{P}^5$? The answer should be 89. I apologize again that if the question is too…
user148177
  • 1,315
4
votes
0 answers

Differentials on a curve

Say I have an algebraic curve $C$ over a field $k$ and a group $G$ acting on $C$. Under what conditions on $C$ and/or the action of $G$ on $C$ can one conclude that $H^0(C,\Omega^1_C)^G = H^0(C/G,\Omega^1_{C/G})$ holds? For example, if the morphism…
Evariste
  • 2,620
4
votes
0 answers

Covering a closed disk in a rigid analytic space by residue classes

Recently I have been reading through the PhD thesis of Dr. Louis Brewis, "Ramification theory of the p-adic open disc and the lifting problem", which is available free…
Garnet
  • 986
4
votes
2 answers

Reid's UAG problem 4.7: isomorphism of affine line with a curve

Let $C:$ $(Y^2=X^3+X^2)\subset \mathbb{A}^2$; the familiar parametrization $$ \varphi\colon \mathbb{A}^1 \to C,$$ given by $$ T \mapsto (T^2-1,T^3 -T)$$ is a polynomial map, but is not an isomorphism (why not?). Find out whether the…
Teddy
  • 2,396
4
votes
0 answers

Why does $H^1(X,\mathbb{Z})$ span a full lattice in $H^1(X,O_X)$ if $X$ is kahler?

We have the exponential short exact sequence for compact complex manifolds. Why does the image of $H^1(X,\mathbb{Z})$ span $H^1(X,O_X)$ over $\mathbb{R}$ if $X$ is kahler? The map is injective, I was wondering why the case can't happen: (for…
user93417
4
votes
2 answers

Notation: Tensor of a sheaf and residue field

This is a simple question of notation. Let $k(p)$ be the residue field of a point $p$ on $\mathbb{P}^{N}$. How is defined (and where is) the sheaf $\mathcal{O}_{\mathbb{P}^{N}}(1) \otimes k(p)$? Also, what could mean the tensor…
rla
  • 2,345
4
votes
1 answer

Showing that a power of an ample sheaf is equivalent to an effective Cartier divisor

I am trying the following exercise: Let $X$ be a quasi-projective scheme over a Noetherian ring A. Let $\mathcal{L}$ be an ample sheaf on $X$. Show that there exists an $m \geq 1$ such that $\mathcal{L}^{\otimes m} \cong \mathcal{O}_X(D)$ for some…
user101036
  • 818
4
votes
1 answer

Connection between degree of dominant morphism and cardinality of fibres

My exercise: Let $f:X\rightarrow Y$ be a dominant morphism of curves. For any dominant morphism, the degree of it is defined to be $[K(X):K(Y)]$ with $K(Y)$ identified with $f^*(K(Y))$. Prove that the fibres of $f$ have at most $deg(f)$ points if…
bbnkttp
  • 1,909
4
votes
2 answers

Conditions under which a bijective morphism of quasi-projective varieties is an isomorphism?

I'm reading a paper by Nakajima (Quiver Varieties and Tensor Products), and I'm having a hard time understanding his proof of Lemma 3.2. Essentially, we have two (quasi-projective) varieties, say $X$ and $Y$, that we would like to show are…
Joel
  • 198
4
votes
0 answers

Cohomology for line bundles on $SL_3/B$ in characteristic $p$

Cohomology of line bundles on the flag variety $SL_3/B$ can be computed using the Bott-Borel-Weil formula in the case the ground field has characteristic zero. In this way one obtains an explicit formula for the dimensions of the cohomology…
Bonanza
  • 1,203
1 2 3
…
99 100