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
5
votes
1 answer

An automorphism acting on the sheaf of differentials

I am trying to do the following problem, taken from Iitaka's "Algebraic Geometry". Let C be a smooth, geometrically connected curve of genus g over a field k. Assume that $g \geq 2$. If $f \in Aut(C)$ satisfies that $f^\ast w = w$ for all $w \in…
TottenLeiber
  • 51
5
votes
3 answers

Are there non-continuable functions that become continuable when raised to some power?

Let $X$ be a complex algebraic variety (integral, separated scheme of finite type over $\mathbb C$) and $U\hookrightarrow X$ an open subvariety. I will say that $f\in\mathscr O_X(U)$ is continuable if there exists some $g\in \mathscr O_X(X)$ with…
user38451
5
votes
1 answer

fiber factors of affine schemes

Let $X,Y \neq \emptyset$ be sufficiently nice schemes over, say, a field $k$. Assume that $X \times_k Y$ is affine. Does it follow that $X$ and $Y$ are affine? Perhaps this works with Serre's criterion.
Martin Brandenburg
  • 163,620
5
votes
0 answers

Rational Elliptic Fibration from Pencil of Cubics

A rational elliptic fibration can be obtained from the total space of a pencil of cubics (i.e. $\mathbb{P}^2$), by Blp the 9 basepoints of the pencil. If the basepoints are distinct, the resulting elliptic fibration has 12 singular, nodal fibers.…
Caliper
  • 158
  • 7
5
votes
1 answer

Degree of effective Cartier divisor

Following http://www.math.columbia.edu/~masdeu/files/notes/ModForms.pdf, define an effective Cartier divisor of an $S$-scheme $f: X \rightarrow S$ as a closed subscheme $Z \subseteq X$, such that $Z$ is flat and finite over $S$ via $f$. $D$ defines…
Cocopuffs
  • 10,307
5
votes
1 answer

Self-intersection of Canonical divisors on Ruled Surface

Everything on $\Bbb{C}$. Let $S$ be ruled surface over a curve $C$ and $K$ a canonical divisor on $S$. What is $K^2$ ? In particular I would like to understand if $K^2=0$ when $C$ is either rational or elliptic. Thanks.
Heitor Fontana
  • 3,676
5
votes
1 answer

Natural description of fibers of blowup of projective space along a subspace

Let $\mathbb{P}^n$ be a projective space, and let $\mathbb{P}^k$ a linear subspace. There are many descriptions of $Bl_{\mathbb{P}^k}\mathbb{P}^n$, but I haven't seen one that's really intrinsic, they tend to rely on choosing a subspace…
Charles Siegel
  • 2,157
5
votes
1 answer

Locally finite + quasi-compact versus finite

Let $f: X \rightarrow Y$ be a morphism of, say, locally noetherian schemes. Suppose that $f$ is locally finite (that is for every open affine subset $U= spec A$ of $Y$, $f^{-1}(U)$ can be covered by open affine subsets $V_i=spec B_i$ such that each…
Visitor
  • 787
5
votes
1 answer

Find the ideal corresponding to an affine variety

Suppose $X \subseteq \mathbb{A}^2$ is defined by the equations $f: x^2 + y^2 =1$ and $g: x= 1$. Find the ideal $I_X$ of all the regular functions that vanish on $X$. Is it true that $I_X= (f,g)$? My attempt : The only common solution of the…
cip
  • 1,127
5
votes
1 answer

Inverse image of Veronese like map

The following is a very elementary question but I can't find the error: Denote $D$ the diagonal $\{[x_0:x_1],[x_0:x_1]\} \subset \mathbb P^1$x $\mathbb P^1$. Let $\phi: \mathbb P^1 \longrightarrow \mathbb P^2$ defined by …
M. E.
  • 348
5
votes
1 answer

A question on torsion sheaves

Im not sure if Ive got this right: Let X be an integral scheme and $\mathcal{F}$ a coherent sheaf. Then $\mathcal{F}$ is torsion if and only if it is not supported at the generic point. It is is easy to see that if the stalk vanishes, any open…
Louis
  • 116
5
votes
0 answers

Is there a noetherification of locally noetherian schemes?

I'm curious about the following: Is there a noetherification of locally noetherian schemes? My motivation is the $1$-point compactification of locally compact hausdorff spaces. Of course schemes are very rigid, so I wouldn't expect it to exist as…
Eivind Dahl
  • 1,657
5
votes
1 answer

What are the equations for the image of an algebraically defined subset under the Segre embedding?

Let $\psi: \mathbb{P}^r \times \mathbb{P}^s \to \mathbb{P}^N$ be the Segre embedding with $N = rs + r + s$, as in Hartshorne exercise I.2.14. To be explicit: the image of the pair $([a_0 : \ldots : a_r], [b_0 : \ldots : b_s])$ is $[\ldots : a_ib_j :…
Ricardo Buring
  • 6,323
5
votes
0 answers

All homogeneous rings of a projective variety

This question is related to this one. What I am wondering is if I have a graded ring $A$ such that $X=Proj(A)$, then is there some specific classification of $B$'s such that $X=Proj(B)$. I am particularly interested in the case where…
BBischof
  • 5,807
5
votes
1 answer

A morphism that is defined over $\mathbb{Q}$ and is an isomorphism over $\mathbb{C}$ may not be an isomorphism over $\mathbb{Q}$.

Is there an example of two algebraic varieties $X,Y$ over $\mathbb{Q}$ and a morphism $f:X\rightarrow Y$ defined over $\mathbb{Q}$, that is an isomorphism over $\mathbb{C}$ but not over $\mathbb{Q}$? That is, the inverse $f^{-1}$ doesn't have…
nowhere dense
  • 2,816
1 2 3
…
99 100