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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an irreducible quadric $ X \subset \Bbb A^n$ d is birational to some $\Bbb A^m$

I want to prove that an irreducible quadric $ X \subset \Bbb A^n$ defined by a quadratic equation $ F(T_1,\ldots,T_n)=0$ is rational (i.e birational to some affine space $\Bbb A^m$ ). I'm not sure ...
3
votes
2answers
237 views

Intersection of quasi-compact open subsets of an affine scheme

Let $X = \mathrm{Spec}(A)$ be an affine scheme. Let $U$ be a quasi-compact open subset of $X$. Then there exist an affine scheme $Y$ and a morphism $f\colon Y \rightarrow X$ such that $f(Y) = U$, ...
3
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1answer
86 views

Associated points of a scheme are contained in an open subset

Recall that we define the set of associated points of a locally Noetherian scheme $X$ as $\operatorname{Ass}(\mathcal{O}_X) = \{ x \in X : \mathfrak{m}_x \in ...
3
votes
1answer
130 views

Diophantine equations and Groebner bases

I'm trying to teach myself the basics of algebraic geometry and have run into something that I don't understand. I know that the problem of deciding whether a Diophantine equation $P(\vec{x}) = 0$ ...
3
votes
1answer
211 views

Sheaf of Relative Differentials for Curves (Hartshorne)

I am trying to understand the sheaf of relative differentials for the case of nonsingular curves. Let's use Hartshorne as a reference, thus a curve is an integral scheme of dimension 1, proper over ...
3
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1answer
480 views

Definition of the Ideal Sheaf

Let $Y$ be a closed subscheme of a scheme $X$ and let $i:Y \rightarrow X$ be the inclusion morphism. Then the ideal sheaf of $Y$ is defined to be the kernel of the morphism of sheafs $i^{\#}: ...
3
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1answer
175 views

the elliptic curves with j-invariant zero

Let $B\in K^\ast$, where $K$ is a number field. Let $y^2=X^3+B$ be the Weierstrass equation for an elliptic curve $E_B$ over $K$. Note that the $j$-invariant of $E$ is zero. When is $E_B$ ...
3
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1answer
121 views

etale fundamental group of a product

Let $X,Y$ be noetherian connected schemes over an algebraically closed field $k$ and let $\overline{x},\overline{y}$ be geometric points on them. There is a canonical homomorphism $\pi_1(X \times ...
3
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1answer
192 views

Morphism of algebraic varieties with vanishing differential

In differential geometry, we know that given a smooth map between smooth manifolds $\phi:X\to Y$, such that $X$ is connected and $d\phi|_x\equiv0$ for all $x\in X$, then $\phi$ is constant (this ...
3
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1answer
79 views

When is this quotient by an action on the product of a variety with itself non-singular

Let $X$ be a smooth projective geometrically connected variety over a field $k$. Let the cyclic group $G=\{e,a\}$ with two elements act on $X \times X$ via $a\cdot (x_1,x_2) = (x_2,x_1)$. When is ...
3
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1answer
182 views

Restriction of flat morphism

Suppose that $f\colon X\to Y$ is a flat morphism of varieties over an algebraically closed field $k$. Let $E\subseteq X$ and $F\subseteq Y$ be closed subvarieties such that $f(E) = F$. Is it true that ...
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1answer
272 views

Tangent space of a projective variety is well-defined

For $X$ an affine variety and $p \in X$ define $T_p X = \mathrm{Der}(k[X], \mathrm{ev}_p)$. Claim: If $Y = X \setminus Z(f)$ is some Zariski open affine subvariety of $X$ and $p \in Y$, then $T_p X ...
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votes
2answers
185 views

How to prove $\mathcal{l}(D+P) \leq \mathcal l{(D)} + 1$

Let $X$ be an irreducible curve, and define $\mathcal{L}(D)$ as usual for $D \in \mathrm{Div}(X)$. Define $l(D) = \mathrm{dim} \ \mathcal{L}(D)$. I'd like to show that for any divisor $D$ and point ...
3
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2answers
190 views

Quotient of an affine variety by a finite group coincides with topological quotient as a point set?

I have just read the construction of the quotient of a closed subset $X$ of affine space by a finite group $G$ of automorphisms of $X$, in Shafarevich, Basic Algebraic Geometry I. Shafarevich gives ...
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votes
2answers
517 views

Unions and intersections of algebraic varieties

Let A = $k[x_1, x_2, \ldots, x_n]$ and let $I_{\lambda}$ be an ideal of A. Let J = $\sum_{\lambda \in \Lambda} I_{\lambda}$ be a finite sum. Show that $V(J) = \cap_{\lambda \in \Lambda} ...
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1answer
135 views

Question concerning the Hodge conjecture.

Let $X$ be a projective complex manifold of (complex) dimension $n$. Let $A \subset X$ be a closed submanifold and $[A]$ be the Poincare dual to its fundamental class. Can you please answer the ...
3
votes
1answer
311 views

Fiber product of sheaves

If one has a topological space $X$ and three presheaves resp. sheaves $F$ and $G$ and $H$ of abelian groups on it with morphisms of presheaves resp. sheaves $F\rightarrow H$, $G \rightarrow H$, then I ...
3
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1answer
186 views

Characterization of Horizontal Irreducible Divisors

I´m asking for a proof of a fact used by Arakelov in his paper: Intersection Theory of Divisors on an Arithmetic Surface (page 1169 row 16). He gives no references or explanations for this fact. The ...
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votes
1answer
159 views

An exercise from the Harris's book

I need a hint on an exercise: Let $K$ be an algebraically closed field. Prove that any finite set $\Gamma \subset KP^n$ such that not all of its points lie on the same line can be given by ...
3
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1answer
118 views

étale fundamental group of strictly henselian discrete valuation ring minus closed point

Let $A$ be a strictly henselian discrete valuation ring. What is $\pi_1(\operatorname{Spec}(A) \setminus \{s\})$? I thought it is a semidirect product of a pro-$p$-group (the wild ramification group) ...
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2answers
533 views

Intuition for blowing-up and the Rees algebra

Starting from an informal understanding of blowing-up as replacing a subscheme by the possible directions into it (or some more accurate formulation of this), how does one justify the definition of ...
3
votes
3answers
223 views

a counterexample for the morphism of sheaves

Suppose $\textbf{F}$ and $\textbf{G}$ are two presheaves over a topological space $X$,and $\mu:\textbf{F}\longrightarrow \textbf{G}$ is a morphism of presheaves which is surjective.We have a naturally ...
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1answer
233 views

Finding a pencil of elliptic curves parametrized by a given modular surface

The following is an attempt to formulate a couple of questions which have been lurking in the back of my mind for a while. I'm sorry if this is long, or if my terminology is not correct, or if my ...
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1answer
163 views

Question on schemes and function fields

Let $X=Spec(R)$ be an irreducible noetherian scheme and $\eta$ the unique minimal prime ideal of $R$. Let $U$ and $V$ be open sets in $X$ and $p$ a point with $p\in V\subseteq U\subseteq X$. If $X$ ...
3
votes
1answer
138 views

Projective, Quasi-Projective

X is a projective variety, W is a quasi-projective variety over the algebraically closed field k. I would like to construct a k-algebra isomorphism between O(XxW) and O(W) (the rings of regular ...
3
votes
1answer
112 views

Toric fano threefolds

According to Batyrev, there are exactly 18 types of smooth toric Fano threefolds. As projective toric varieties, these are defined by the normal fan of certain 3-dimensional polytopes. Does anyone ...
3
votes
1answer
432 views

Morphisms of finite type are stable under base change

I am trying to prove that morphisms of finite type are stable under base change, but I am having some trouble moving from the case where everything is affine to the general case. Suppose $f:X ...
3
votes
1answer
290 views

Connected components of a fiber product of schemes

The underlying set of the product $X \times Y$ of two schemes is by no means the set-theoretic product of the underlying sets of $X$ and $Y$. Although I am happy with the abstract definition of fiber ...
3
votes
1answer
238 views

the sheafification of a constant presheaf

Let X be a topological space and A be an abelian group. Give A the discrete topology. For any open set U of X, Let $\cal A(U)$ be the group of all continuous aps of U into A. Thus with the usual ...
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2answers
208 views

Why should one expect valuations to be related to primes? How to treat an infinite place algebraically?

I understand the mechanics of the proof of Ostrowski's Theorem, but I'm a little unclear on why one should expect valuations to be related to primes. Is this a special property of number fields and ...
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1answer
47 views

Calculating canonical divisor in product of projective spaces.

Let $X$ be an intersection of two divisors of bidegree $(a,b)$ and $(c,d)$ in $\mathbb{P^2}\times \mathbb{P^2}$. Then how can I find the canonical divisor $K_X$? I'm asking because I have no ...
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votes
1answer
48 views

The local rings of $xy=0$ and $xy+x^3+y^3=0$ are not isomorphic, but have isomorphic completions?

I know that if you have a commutative local ring $R$, and you take its completion $\widehat{R}$ the inverse limit of the $R/\mathfrak{m}^i$, you get another local ring. However, nonisomorphic local ...
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votes
1answer
62 views

Calculating of genus of a curve

Let $C$ be a curve over $\mathbb{F}_q$ in projective plane. So $C$ can be done as zeroes of some gomogeneous polynomial $\in \mathbb{F}_q[x,y,z]$ with degree $n$. Whether is there algorithm which is ...
3
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1answer
77 views

Understanding the stack $B\mathbb{Z}$

Here, let $\mathbb{Z}$ be the group scheme whose functor of points is the constant functor which takes a connected affine scheme to the group $\mathbb{Z}$. I'm having a bit of trouble understanding ...
3
votes
1answer
80 views

What is GAGA for dimension 1 ? (Historical Question)

I know Riemann surfaces are actually algebraic curves, i.e. all Riemann surfaces can be simply embedded into some projective space $\mathbb{P}^n$. But this doesn't indicate me more correspondences ...
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1answer
106 views

How to prove that $CP^4$ cannot be immersed in $R^{11}$

Please let me know how to prove that $CP^4$ cannot be immersed in $R^{11}$. I know a proof using an integrality theorem for differentiable manifolds but I want to know if a more direct and simple ...
3
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1answer
49 views

Invariants of $O(2) \times O(2)$ under simultaneous conjugation

Let $G= \textrm{O}(2)$ be the group of orthogonal $2 \times 2$ matrices over $\mathbb{C}$. $G$ acts on $G \times G$ by conjugation: $g \cdot (a,b) :=(g a g^{T}, g b g^T)$. This induces an action on ...
3
votes
1answer
48 views

Blow-ups followed by contractions

Let $S$ be a minimal, non-singular complex projective surface. $\widehat S$ is the surface obtained by $r$ blow-ups of $S$ at the points $x_1,\ldots,x_r\in S$. Clearly $\widehat S$ contains exactly ...
3
votes
1answer
54 views

If the coordinate ring $k[V]\simeq k$, is $V$ necessarily a singleton?

Suppose $V$ is a quasi-affine set such that $k[V]\simeq k$, where $k[V]$ is the coordinate ring, and $k$ a field such that $k=\bar{k}$. Does this force $V$ to just be a point? I'm curious because I ...
3
votes
1answer
44 views

Questions about tangent and cotangent bundle on schemes

In differential geometry, for a smooth manifold $M$ we have the definition of the tangent bundle and the cotangent bundle and then $k$-forms are defined to be (smooth) sections of the $k$-exterior ...
3
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1answer
65 views

If $f_1,…,f_{n+1}\in\mathbb{C}[x_1,…,x_n]$, is there a polynomial in the coefficients which vanishes iff the $f_i$ have a common root?

My question is as in the title: Suppose $f_1,...,f_{n+1}\in \mathbb{C}[x_1,....,x_n]$. Is there polynomial $g$ (or a system of polynomials) with variables given by the coefficients of the $f_i$ ...
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2answers
72 views

T//W for adjoint type group PGL3

Let $G$ be a reductive algebraic group and $T$ a maximal torus (over $\mathbb{C}$). It is well known that if $G$ is simply connected type then $T//W = \mathbb{A}^r$. I want to verify that the ...
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1answer
71 views

Hartshorne Exercise III 6.2 (a)

Let $X=\mathbb{P}^1_k$, with $k$ an infinite field. Show there does not exist a projective object $\mathcal{P}\to\mathcal{O}_X\to 0$. The author suggests to consider surjections of the form ...
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1answer
75 views

Cohomology Calculation

A couple of days ago I asked this Question on calculating hypercohomology I tried a similar example for $(\mathbb{C}^*)^2$, and I have a couple of questions. Here is my calculation: We have a ...
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1answer
52 views

Kernels of power surjective maps

Suppose $k$ is an algebraically closed field, and $A$ and $B$ are finitely generated, commutative, graded $k$-algebras. Suppose $\varphi:A\to B$ is a map of $k$-algebras. Notice if $B$ is a domain, ...
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2answers
90 views

on the coordinate ring of $\mathbb{A}^n \times \mathbb{P}^{m}$

Consider the product $\mathbb{A}^n \times \mathbb{P}^{m}$. Let $x_i$ be affine coordinates on $\mathbb{A}^n$ and $y_j$ homogeneous coordinates on $\mathbb{P}^{m}$. Question: Is ...
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1answer
42 views

Surjective étale morphisms on points. [closed]

Let $X$ and $Y$ be schemes over a field $K$. We assume, moreover, $X$ and $Y$ to be of finite type, separated and geometrically integral. Let $f:X \rightarrow Y$ be a surjective étale morphism. Is it ...
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1answer
125 views

Learning roadmap for classical algebraic geometry (italian school)

Can someone suggest a learning roadmap for classical algebraic geometry as develop by the great italian school? Severi has a few books but in italian. I would like to know what are the best English ...
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1answer
84 views

Picard group of a smooth projective curve

I have two (related) questions regarding the Picard group: 1) Are there examples of smooth projective curves with large Picard groups (say $Pic(X)\simeq\mathbb{Z}^n)$ for any $n$)? 2) In general, ...
3
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1answer
64 views

Subtle aspect of closed subscheme

Let me define a closed subscheme of a scheme $X$ as: An equivalence class of data in the form $$(Z,Y,i,i^\sharp)$$ where $Z$ is a closed subset of $X$, $(i,i^\sharp): Y \to X$ is a morphism of ...