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. ...

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16 views

a question regarding some proof in Klaus Hulek algebraic geometry

I have a problem with the proof of proposition 1.45. This is the p.42 of Klaus Hulek's Elementary Algebraic Geometry. From the last equation, the author concludes that f(V) is contained in W. But I ...
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0answers
18 views

Common zeros and GCD of polynomials

Facing another algebraic geometry problem: Let $p,q \in T[x,y]$. Prove the set $V(p,q)$ is finite if and only if set $V(GCD(p,q))$ is finite. ($V(p)$ of course meaning the subset of $A^2(T)$ where p ...
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1answer
9 views

a question regarding the definition of localization of ring

Let S be a mutliplicative system. Then localization of R is defined by an equivalence relation on R x S. The relation is (a,b) ~ (c,d) if there is an s in S such s(ad-bc)=0 Regarding this, I can't ...
2
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0answers
32 views

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 ...
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1answer
22 views

Acyclic but not flasque sheaf of abelian group?

Is there a sheaf of abelian groups which is acyclic but not flasque? Maybe we can try $0\to \mathcal{F'}\to \mathcal{F}\to \mathcal{F''}\to 0$ where $\mathcal{F',F''}$ are flasque but $\mathcal{F}$ ...
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0answers
35 views

Consequence of the Hodge index theorem

I'm trying to find an example of two divisors $D_{1} \ D_{2}$ on a complex algebraic projective surface $S$ such that: $D_{1}\equiv D_{2}$ where the equivalence relation is the numerical ...
3
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0answers
51 views

Application of the Riemann-Roch theorem

If C is a quadric hyperelliptic curve ($g(C)=3$ and the canonical line bundle is very ample) contained in the two dimensional complex projective space and $K_C$ is the canonical line bundle of ...
2
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0answers
19 views

Different definitions of an affine algebraic set

Hartshorne assumes we are working over an algebraically closed field $k$ and we simply define algebraic sets to be any set of the form $V(I)\subset k^n$, where $I\subset k[X_1,\ldots,X_n]$ is an ...
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3answers
67 views

stalk of projective variety in terms of the coordinate ring

Let $X \subseteq \mathbb{P}^n$ be an embedded projective variety over some field $k$ with its corresponding homogeneous coordinate ring $R = k[X_0,\dots,X_n]/I(X)$. Let further $X = \bigcup_{i=0}^n ...
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2answers
23 views

Hypersurface in $\mathbb P^n$ containing a linear subspace of dimension $r \geq n/2$ has singular points

I'm trying to prove that if I have a hypersurface $X = Z(F)$ (where $F \in K[x_0, \dots, x_n]_{d>1}$) which contains a linear subspace of dimension $r \geq n/2$ then there exists singular points on ...
2
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0answers
33 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 ...
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1answer
18 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
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1answer
41 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 ...
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1answer
35 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 ...
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0answers
36 views
+50

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 ...
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1answer
21 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 ...
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0answers
49 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 ...
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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 ...
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1answer
42 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
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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
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2answers
67 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 ...
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1answer
35 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 ...
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0answers
38 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
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0answers
38 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 ...
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0answers
25 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 ...
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0answers
14 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 ...
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0answers
15 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)^{'} ...
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0answers
29 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, ...
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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 ...
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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
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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 = ...
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0answers
26 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 ...
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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) ...
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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 $\Bbb P^n$, ...
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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 ...
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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
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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 ...
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0answers
14 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 ...
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0answers
62 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 ...
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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 ...
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1answer
21 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 $A^2( \mathbb{C})$. ...
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1answer
50 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 ...
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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
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1answer
5 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
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1answer
52 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 ...
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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 ...
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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 ...
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1answer
54 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 ...
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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) = ...
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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 ...