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
7
votes
1 answer

Projective varieties - basics

I am taking an introductive course in (real) algebraic geometry and I got stuck at some basic exercises. They regard affine and (real) projective varieties, as follows: Prove that the punctured projective space, $\mathbb{P}^n - \{x\}$ is neither…
Adrian Manea
  • 1,145
6
votes
2 answers

The second and third Chern classes of Calabi-Yau threefolds

Let $X$ be a smooth projective Calabi-Yau threefold. Then the first Chern class vanishes: $$c_1(X)=c_1(T_X)=0.$$ Is anything known about $c_2(X)$ and $c_3(X)$? What about $c_2$ of a K$3$ surface? (I am sorry if this is very well-known. This…
Brenin
  • 14,072
6
votes
2 answers

Genus of intersection of two surfaces in $\mathbb{P}^3$

Let $F_1$ and $F_2$ be two (smooth) surfaces in $\mathbb{P}^3$, of degrees $d_1$ and $d_2$ respectively. Let $C$ denote curve given as their intersection. How one can compute arithmetical genus of the curve $C$? Perhaps I need to add some…
Alex
  • 6,264
6
votes
1 answer

rational normal curve of degree 3 not written by intersection of two quadrics

I'm learning about rational normal curves of degree n. And the book says that rational normal curves of degree 3 cannot be written by intersection of two quadrics. I can visualize the situation in my head, but cannot formulate a rigorous…
Keith
  • 7,673
6
votes
0 answers

A smooth cubic is not rational

We consider projective curves over the closed field $\mathbb{k}$. It can be proven that the curve is rational iff its genus $g=0$. Also the curve is birationally equivalent to a nonsigular cubic iff its genus $g=1$. This means, in particular, that…
Mikolay
  • 215
6
votes
1 answer

Generators of the ring of sections of a line bundle

This is probably a really simple question, but I never really studied classical algebraic geometry, and so I don't know where to start looking. Suppose $X$ is a smooth projective algebraic curve over a field $k$, and $\mathcal{L}$ a line bundle on…
David Loeffler
  • 9,584
6
votes
2 answers

Rational Points, classical versus modern notion

In classical algebraic geometry, a $\mathbb Q$-rational point on a, say, complex affine variety $V\subseteq\mathbb C^n$ is a point $p=(p_1,\ldots,p_n)$ with $\forall i: p_i\in\mathbb Q$. Now, in modern language, a $\mathbb Q$-rational point is a…
Jesko Hüttenhain
  • 14,792
6
votes
1 answer

Formal schemes vs formal power series

Take $X = \mathbb{A}^1$ and $Y = \{0\}$. I want to take the formal group scheme at $Y \subset X$. This is a locally ringed space, $(Y, \mathcal{O}_{ \hat{X}})$ where $\mathcal{O}_{\hat{X}}$ is the $(x)$-adic completion of $k[x]$, i.e.…
user148177
  • 1,315
6
votes
0 answers

Homs of abelian varieties and Tate/Dieudonne modules

I have basic question: Suppose I have two abelian varieties $A,B$ over $\overline{\mathbb{F}_p}$. Let $M(A), M(B)$ denote the (covariant) Dieudonne modules (over $W$ the Witt vectors of $\overline{\mathbb{F}_p}$ ) of their respective p-divisible…
qewrrewer
  • 61
  • 1
6
votes
3 answers

Morphism of schemes $f\colon X\to Y$ associated to a continuous map of the underlying spaces $|X|\to |Y|$

I am sorry for asking two questions in one but they are strongly related. What is an example of (affine?) schemes $X=(|X|,\mathcal{O}_X)$ and $Y=(|Y|,\mathcal{O}_Y)$ and a map of topological spaces $|f|\colon|X|\to |Y|$ that cannot be promoted…
user8463524
  • 2,157
6
votes
1 answer

scheme-theoretic image behaves nicely with composition, base change?

Scheme-theoretic image is still somewhat of a mystery to me, and I wasn't able to work out proofs of either of the following two statements that seem plausible to me: If $X\to Y\to Z$ is a map of schemes, $Y'\subset Y$ is the scheme-theoretic image…
LCL
  • 1,185
  • 7
  • 22
6
votes
0 answers

Theorem 8.17 , Chapter II, Hartshorne

Let X be a nonsingular variety of dim n over an algebraically closed field k. Let Y be an irreducible closed subscheme defined by a sheaf of ideals $\mathscr I$. Then I want to prove that Y is a nonsingular variety over k iff (1) $ \Omega_{Y/k} $…
Suhas
  • 497
6
votes
1 answer

Closed orbit of $\operatorname{PGL_2} K$ acting on $\mathbb{P}(\operatorname{Sym}^4 V)$, $\operatorname{dim} V = 2$

The standard action of $\operatorname{GL_2} K$ on $V$ induces an action of $\operatorname{PGL_2} K$ on $\mathbb{P}(\operatorname{Sym}^4 V)$. So far, I understood how all the orbits can be obtained via the $j$-function, but I'm struggling to see some…
Paul
  • 1,232
6
votes
1 answer

Why the ideal defining a closed subscheme is unique?

Let $X$ be a scheme. We defined a closed subscheme of $X$ to be a scheme $(Z, \mathcal{O}_Z)$ such that $Z$ is a closed subset of $X$ and $i_*\mathcal{O}_Z \simeq \mathcal{O}_X/\mathcal{J}$, where $\mathcal{J}$ is an ideal of $\mathcal{O}_X$ and $i…
Eskil
  • 431
6
votes
1 answer

Degree 3 algebraic curve with a triple point

The following problem appeared on an exam I had yesterday. I was unable to solve it, but I would like to know the solution. Let $k$ be algebraically closed, and let $f\in k[X,Y]$ be a polynomial with $\deg f = 3$. Assume that $V(f)\subseteq…
Espen Nielsen
  • 3,863
  • 17
  • 31
1 2 3
…
99 100