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. ...

learn more… | top users | synonyms (1)

1
vote
0answers
23 views

Category of schemes with flat morphisms

Consider the category whose objects are schemes, and for every two schemes $X$ and $Y$, morphisms $\operatorname{Hom}(X,Y)$ consist of flat morphisms $X\rightarrow Y$, only. Does this category have a ...
1
vote
1answer
14 views

non-singular Riemann surface implies irreducible polynomial without connectedness?

Let $$ F(w,z) = \sum_{i=0}^n a_i(z)w^{n-i}$$ be a polynomial in $z,w$. Define a Riemann surface as the set $$\Gamma:= \left\{ (z,w)\in \mathbb C^2 \mid F(z,w)=0 \right\} $$ and call it non-singular if ...
2
votes
1answer
31 views

Restriction of sheaf via inclusion induces isomorphism on stalks

Let $i: Z\rightarrow X$ be the inclusion of $Z $ as a subspace of $X $. Let $\mathscr{F}$ be a sheaf on $X$. The restriction of $\mathscr{F}$ to $ Z $ is defined as the sheafification of $U\mapsto ...
0
votes
0answers
18 views

Defining the map between tangent space in locally ringed space

I had a doubt studying locally ringed space about what is the canonical map between tangent spaces in the case the residue field is different: Let $(f,f^*):(X,O_X) \to (Y,O_Y)$ a morphism of locally ...
0
votes
0answers
24 views

A possible mistake in Hartshorne chapter 2 proposition 2.6

Here is the context of this question. Hartshorne claim that $O_X(U)\cong \beta_*(O_V)(U)=O_V(\beta^{-1}(U))$ for any open $U\subset X=\operatorname{Spec}A$,but it is possible that ...
1
vote
1answer
13 views

Index intersection of ample divisors

I'm trying to prove that the sum of two ample divisors on a projective complex algebraic surface S is it self an ample divisor. To do this i need to verify that the index intersection between two ...
2
votes
0answers
39 views

Is the maximal ideal of a localization at a prime ideal principal?

Let $X$ be a closed subvariety of $\mathbf P^{n}_{k}$ which is nonsingular in codimension one. Let $Y$ be a subvariety of $X$ of codimension one, let $\eta$ be its generic point. First question: is ...
0
votes
0answers
27 views

$H^1(\mathbf{F}_q, Pic(\bar{X})) = 0$?

Let $X/\mathbf{F}_q$ be a smooth projective geometrically integral variety. Does it follow that $H^1(\mathbf{F}_q, Pic(\bar{X})) = 0$? Isn't this just Lang-Steinberg? I think Lang-Steinberg gives us ...
1
vote
1answer
41 views

Lefschetz hypersurface theorem

Let $X \subseteq \mathbf{P}^4$ be a hypersurface over $k$ algebraically closed. Why do we have $Pic(X) = Pic(\mathbf{P}^4)$ by the Lefschetz hypersurface theorem? I only see for $\mu_n$ coefficients ...
2
votes
1answer
44 views

Geometric interpretation of $H^1_{Zar}(X,\mathcal O_X)$

Let $X/k$ be a smooth projective geometrically integral variety, perhaps over $k$ algebraically closed. What is the geometric interpretation of $H^1_{Zar}(X,\mathcal{O}_X)$? Does it have something to ...
3
votes
2answers
59 views

Elementary algebraic geometry

Let $p(z,w)=z^2+w^2-zw+1,$and $Z(p)=\{(z,w)\in\mathbb{C}\times\mathbb{C}|\,p(z,w)=0\}.$ Is this variety irreducible? Is $Z(p)$ a connected subset of $\mathbb{C}\times\mathbb{C}$ ? (in usual topology ...
2
votes
1answer
32 views

Euler characteristic of the structure sheaf

I started to study vector bundles on the spaces, then I have my first contact with the instanton bundles, (bundles that are cohomology of the $0 \to \mathcal{O}^{k}_{\mathbb{P}^n}(-1) \to ...
1
vote
0answers
35 views

Cohomology class with trivial restriction to a very general fiber

Let $f:X\to S$ be a flat morphism of smooth complex projective varieties. Let $s\in S$ be a very general point. Suppose that $\omega\in H^{p,p}(X)$ is a cohomology class such that $\omega|_{X_s}=0\in ...
2
votes
0answers
36 views

Where do divisors on curves come from?

I've been learning about Riemann surfaces and the Riemann-Roch theorem, and I've been able to experience some of its great power. However, I'm curious as to what the history behind divisors on curves ...
1
vote
0answers
24 views

On structure sheaf of an affine scheme

I am reading the algebraic geometry notes by Ravi Vakil. When he proves that the structure sheaf on affine scheme is indeed a sheaf (Thm 4.1.2. in his notes), he first proves that it gives a sheaf ...
1
vote
0answers
13 views

What is the relation between modification and blow-up along the base locus?

Let $X \hookrightarrow \mathbb{P}^N$ be a compact sub manifold of dimension $n$. Let $\mathbb{P}(d, N)$ denote the projectivization of degree $d$ homogeneous polynomial on $\mathbb{P}^N$. Each ...
0
votes
0answers
13 views

The normalization of a product of varieties

Let $X,Y$ reduced complex analytic spaces, $X^{'}$ and $Y^{'}$ the normalizations of $X$ and $Y$, respectively. Let $(X \times Y)^{'}$ the normalization of $X \times Y$. Is true that $(X \times Y)^{'} ...
-1
votes
0answers
23 views

Relation between elements of a ring and their annihilators

let $(R.m)$ be a local ring and $x,y$ two elements of $R$ and for ideal $I$ of $R$, we have $x$ is in $I$, $ann(x)=ann(y)$ and $x$ is uniqu minimal ideal of $R$, is there any conditions that implies, ...
0
votes
1answer
39 views

Fibred product of schemes

I'm learning algebraic geometry, and I'm having some difficulties in developing intuition for the fibred product of schemes. I can take schemes to be over a field (but not necessarily separably ...
-1
votes
0answers
16 views

Polynomial equation, tried several times can't find answer [on hold]

Tried several attempts at getting an answer for this. (2m^3+4)^2
0
votes
1answer
27 views

local dimension of irreducible varieties

If $X$ is an irreducible (quasi-)affine variety it is well known that each maximal sequence $C_0 \subsetneq C_1 \subsetneq \dots \subsetneq C_d$ of irreducible closed subsets has the same length $d = ...
0
votes
0answers
25 views

How to efficiently check whether two cubics are equivalent

I have a very long list of cubic polynomials in $N$ variables, with $N$ ranging from $2$ to $19$. For my purposes, any two cubics which are related by a rational change of basis in the $N$ variables ...
0
votes
0answers
27 views

Existence of a polynomial 1-17 of Fulton (algebraic curves)

a) Let $ V $ an algebraic set $ A ^ n (K) $ and $ a \ in {A ^ n (K)} $ a point that does not belong to $ V $. Prove that there exists a polynomial $ p \ in {K [x_1, ..., x_n]} $ such that [tex] p (b) ...
-1
votes
0answers
16 views

Let Sn={$x\in Rn+1$; $\langle\ x,x\rangle$=1} a sphere n-dimensional. [on hold]

I study Metric Spaces and I have this problem. Let $\Bbb S^n=\{x\in \Bbb R^{n+1}; \langle x,x\rangle=1\}$ be a $n$-dimensional sphere. The projective space with dimension $n$ is the set $\Bbb P^n$, ...
0
votes
0answers
28 views

$n$-connectivity of principal $G$-bundle

Page 202 of Switzer's Algebraic Topology: Let $\xi=(E,p,B)$ be a principal $G$-bundle such that $E$ is $n$-connected. Then for any pointed CW-complex $(X,x_0)$ of dimension at most $n$, the ...
1
vote
1answer
49 views

short exact sequences of linear algebraic groups and $K$-forms

This is probably a stupid question, but I can't figure it out. Let $G, G', G''$ be linear algebraic groups over a characteristic $0$ field $K$. Say that $G' \in A$ and $G'' \in B$, where $A,B$ are ...
3
votes
1answer
27 views

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 ...
0
votes
0answers
13 views

An isogeny from a split algebraic torus

Suppose that there is an isogeny (in the category of commutative algebraic groups) from a split algebraic torus to a semi-abelian variety. Does it follows that this semi-abelian variety is also an ...
6
votes
0answers
59 views
+100

Why is there no theory of $G$-ic varieties, for linear algebraic groups $G$?

A toric variety is an algebraic variety $X$ with an embedding $T \hookrightarrow X$ of an algebraic torus $T$ as a dense open set, such that $T$ acts on $X$ and the embedding is equivariant. It ...
0
votes
2answers
38 views

Fermat's Curve is not rational (Perrin's “Algebraic Geometry - An Introduction”)

Below is the proof that curve given by $x^n+y^n=1$ for $n\geq 3$, over field of characteristic that does not divide $n$, has no rational parametrization, from Intrduction of Perrin's "Algebraic ...
0
votes
1answer
17 views

Decomposition into irreducible algebraic sets

I am facing following problem and would really appreciate anyone's help: I am to find decomposition of $V(x^2 - y^4, x^3 - xy^2 + x^2y^2 -y^4)$ into irreducible algebraic sets in $A^2( \mathbb{C})$. ...
0
votes
1answer
49 views

Why the unitary group is not a complex algebraic variety?

The question comes from Exercise 1.1.2 of the book "An Invitation to Algebraic Geometry". By definition the unitary group U(n) is the group of all complex matrix that satisfies $U^*U=I$. I know that ...
2
votes
0answers
25 views

Compute with the Tangent Bundle to a projective bundle over projective plane

Suppose I have a vector bundle $E$ on $\mathbb P^2$ and form the projective space bundle $X:=\mathbb P(E)$. Hartshorne (p253) gives an exact sequence to compute $\Omega_{X/\mathbb P^1}$, but I am ...
3
votes
1answer
2 views

Containment of two varieties with a lot of intersection

Given a projective variety $X\subset \mathbb P^n$ and a curve $C\subset \mathbb P^n$, when can I conclude that $C\subset X$, from the fact that $C$ and $X$ have 'many' points in common. I.e., is there ...
3
votes
1answer
51 views

When do functions turn a space into a locally ringed space?

Let $X$ be a topological space, and consider for each open set $U \subseteq X$ a set $F_U$ of functions $U \to k$ into some fixed field $k$. Let $\mathcal{O}$ be the sheaf of $k$-algebras induced by ...
1
vote
1answer
23 views

Generators of ideal of coordinate axes in A^3

So I know that the algebraic set $X$ equal to the union of the three coordinate axes in $\mathbb{A}^3$ is $I(X) = (xy, yz, xz)$ and that this is the fewest number of generators. But it seems that the ...
6
votes
0answers
51 views

“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 ...
1
vote
1answer
53 views

Intersection of two polynomial ideals

In the 4-dimensional affine space $\mathbb{A}^4$ with coordinates $x,y,z,t$, consider $X$ as the union of the planes $$ X'=\{x=y=0\} $$ and $$ X''=\{z=x-t=0\} $$ (I'm working on a algebraically ...
0
votes
0answers
24 views

$\text{div} (\mathcal{L},s) = 0$

I am reading a proposition in Prof. Vakil's notes where he shows that if $X$ is a normal and Noetherian scheme, then $\text{div}$ is injective. He opens by saying that if $\text{div} (\mathcal{L},s) = ...
1
vote
1answer
45 views

degree of morphism of schemes

Let $\phi: Y \to X$ be a finite etale morphism of proper smooth connected schemes over a field $K$ and suppose that the induced morphism $\phi: \overline{Y} \to \overline{X}$ has degree $n$, where ...
1
vote
1answer
45 views

a question regarding Klaus Hulek algebraic geometry

sorry for uploading weird angle pictures..but no other concise ways I can't think of.. this is the p.49 of Klaus Hulek elementary algebraic geoemtry on the last paragraph it says that K[V] is not a ...
0
votes
0answers
29 views

Theorem weak Hilbert

If $ I $ is a proper ideal of $ K [x_1, \ldots, x_n] = F $ then $ V (I) \neq \emptyset$. dm: I want to use that $ J = \langle x_1-a_1, \ldots, a_n x_n \rangle $ is a maximal ideal of $ F $. If I ...
1
vote
0answers
50 views

Does such a polynomial map always exist?

First question: given $$\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}=\frac{P_3(f(x_1,x_2,\dots,x_n))}{P_4(f(x_1,x_2,\dots,x_n))}|\det (J(f(x_1,x_2,\dots,x_n)))|$$ where $P_i$ is ...
3
votes
0answers
34 views

trivialising cover for etale morphisms

Let $f:Y \to X$ be a finite etale morphism of smooth and proper schemes over a field $k$ (not necessarily sep closed). Is there a geometrically connected etale cover $\{U_i\}$ of $X$ which ...
2
votes
1answer
47 views

What is the importance of Jacobian Conjecture and any progress on it?

What is the importance of Jacobian Conjecture?Are there any important central problem with the conjecture as precondition? and any progress on it?
0
votes
2answers
60 views

Algebraic variety in $\mathbb{C^{2}}$

would somebody algebraic-geometry-savvy please help me with this problem, as I am pretty new to algebraic geometry: I have to find smallest algebraic variety (irreducible algebraic set) in ...
-1
votes
0answers
28 views

An Advice Concerning Master's Programme [closed]

Which of these programmes is a better choice, if one wants to pursue a degree in pure mathematics? (In Geometry, Topology and Algebra, in particular, algebraic geometry) 1) ...
0
votes
1answer
17 views

Plotting Particular Conic Section

How would I plot $-2x^2 -2y^2 = 1$ on the x-y plane ? I believe it is an ellipse, since the coefficients have the same sign, I just don't know what the major and minor axes would be nor how to plot.
1
vote
0answers
34 views

Counting the Number of Points in an Algebraic Variety

How can we count the number of points in $$S = \{(x,y) \in \mathbb{Z_m}^2: x^2+ky^2 = c\}$$ where $k,c$ are some positive integers?
0
votes
2answers
26 views

Curve in union of hyperplanes

If a smooth curve $\gamma: [0,T] \to \mathbb R^n$ is contained in the union of hyperplanes $$ \bigcup_{i=1, \dots, N} H_i$$ does it then follow that one can always find time intervals $[t_0, t_1]$ ...