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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73 views

When is diagonal morphism the diagonal map

Suppose $X, Y$ are schemes and $f : X \rightarrow Y$ a morphism and $\pi_1, \pi_2 : X \times_Y X \rightarrow X$ be the two projections and $\Delta : X \rightarrow X \times_Y X$ is diagonal morphism. ...
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57 views

Varieties are isomorphism

Let $f:Y\to X$ be a birational morphism, Y is projective. Let $H$ be a very general ample divisor on $Y$. If $f^{*}f_{*}H=H$, is it true that $Y$ is isomorphic to $X$?
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208 views

What is an “algèbre augmentée sur un corps?” (EGA I)

In EGA I, Chapter 0, (1.1.10), Grothendieck is giving examples of terminal objects in different categories. He says "dans la catégorie des algèbres augmentées sur un corps $K$ (où les morphismes sont ...
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42 views

Connection of $\mathcal{O}(n)$ on a toric manifold

The holomorphic line bundle $\mathcal{O}_X(1)$ over a toric manifold $X$, admits a hermitian connection, $A^{(1)}$, whose $U(1)$ gauge transformation in a local patch of the base space is $$ A^{(1)}...
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1answer
66 views

Doubt in Hartshorne Example 7.17.3, Chapter 2

Let $X$ be a smooth projective variety over $\mathbb{C}$ of dimension $n$. Let $L$ be a line bundle on $X$, and let $V\subset H^0(X,L)$ be a subspace of sections. Suppose that $s_1,..,s_{l+1}$ is a ...
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1answer
48 views

Push-forward of quasi-coherent sheave on affine scheme is quasi-coherent

Let $X=$ Spec$R$, $Y=$ Spec$S$, $f:X \to Y$ be a morphism of schemes. Let $M$ be a $R$-module, and let $\mathcal{F}=\tilde{M}$ be the sheaf on $X$ induced by $M$. How can I show that the pushforward ...
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34 views

How to deduce the usual definition of Quasi-coherent Module over a scheme from the general definition over ringed Spaces

Quasi-Coherent Modules over a Ringed Space : Let $(X,\mathcal O_X)$ be a ringed space. A sheaf of modules $F$ over $(X,\mathcal O_X)$ is called quasi-coherent if for every point $x\in X$ $\exists U\...
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36 views

Group of $\mathfrak a$-torsion points

Silverman defines the Group of $\mathfrak a$-torsion points of an elliptic curve $E/\mathbb C$ (with $\mathfrak a$ an ideal in $\mathrm{End}(E)$) in Advanced topics of elliptic curves as $$E[\mathfrak ...
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1answer
59 views

Do two rational parametric curves intersect only finitely many times?

Suppose there are two rational parametric curves $f = (f_1, \ldots, f_n)$ and $g = (g_1, \ldots, g_n)$ in $\mathbb{R}^n$. I read somewhere that such a parametric expression can always be transformed ...
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32 views

Question about radical ideal

Suppose $\mathbb{R}[X]$ is the normal multivariate polynomial ring where $X = x_1,...x_n$. $\mathbb{R}[X]_t$ is the truncated set such that $\mathbb{R}[X]_t =\left\{f: f \in \mathbb{R}[X], \deg(f) \...
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1answer
55 views

Picard rank of K3s — How can it be small?

I must be missing something. How can the Picard rank of a compact Kahler manifold drop below its Hodge number $H^{1,1}(X)$? For simplicity, let $X$ be a K3 surface. After all, we have the standard ...
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38 views

Comparing morphisms of local nature for equivalent topologies

Let $S$ be a scheme. It is known that the smooth topology on $\textrm{Sch}_S$ is equivalent to the étale topology, basically because every smooth covering can be refined to an étale covering. ...
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88 views

Finite generation of global sections of an invertible sheaf on a quasi-projective scheme

Let $X$ be a projective scheme over a noetherian ring, $\mathcal F$ an invertible sheaf on $X$, and $U$ an arbitrary open subset of $X$. Is $\Gamma(U,\mathcal F)$ a $\Gamma(U,\mathcal O_X)$-module of ...
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43 views

Certain map of modules is iso [Mumford Abelian Varieties]

I have trouble showing the following in the proof of Prop. 2 in Abelian Varieties (pg.70 my edition, Chapter about quotients by finite groups): Suppose you have a Noetherian ring $B=A^G$ as ...
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1answer
63 views

Finite generation of sections of a coherent sheaf

Let $X$ be a quasi-projective scheme over a noetherian ring, $\mathcal F$ a coherent sheaf of $\mathcal O_X$-modules, and $U$ an arbitrary open subset of $X$. Is $\Gamma(U,\mathcal F)$ a $\Gamma(U,\...
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36 views

Topological covers $\mathbb{CP}^1\rightarrow \mathbb{CP}^1$

Suppose we have the $\mathbb{C}$-variety $\mathbb{P}^1_{\mathbb{C}}$. When we analytify this variety we get $\mathbb{CP}^1$. The question is: suppose we have a morphism $f : \mathbb{P}^1\rightarrow \...
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1answer
56 views

How to apply Riemann-Roch Theorem and Kodaira Vanishing Theorem to an ample line bundle?

Let $X$ be an abelian variety. In "Mumford, Abelian Varieties" the Riemann-Roch Theorem has the following form: For all line bundles $\mathcal{L}$ on $X$, if $\mathcal{L}\cong\mathcal{O}_X(D)$, ...
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1answer
39 views

Well-definedness of homogeneous coordinate ring of projective scheme [closed]

Let $I,J \subset S=k[x_0,...,x_n]$ be homogeneous ideals. How can I show that $\operatorname{Proj}S/\bar{I}=\operatorname{Proj}S/\bar{J}$ iff $\bar{I}=\bar{J}$? (Here $\operatorname{Proj}R$ is the ...
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47 views

$\operatorname{Spec}(\mathbb{Z}_p[x])$

I'm trying to solve $\operatorname{Spec}(\mathbb{Z}_p[x]/x^{2015})$, so I need to find the prime ideals of $\operatorname{Spec}(\mathbb{Z}_p[x])$ that contains $x^{2015}$, can anybody help me with ...
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1answer
37 views

Spec($\mathbb{Q}\times\mathbb{Q}$), Spec($\mathbb{Z}\times\mathbb{Z}$)

I know that $\operatorname{Spec}(\mathbb{Z})$ is the set of the zero ideal and the primes, and $\operatorname{Spec}(\mathbb{Q})$ is the ideal zero, because $\mathbb{Q}$ is a field, but it is true that ...
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1answer
42 views

Saturation of homogeneous ideal

Let $I \subset S=k[x_0,...,x_n]$ be a homogenous ideal. The saturation of $I$, $\bar{I}$ is defined to be $\{s \in S: \exists m \; s.t. \; \forall i \; x_i^m s \in I\}$ Is it true that $\bar{I}=(s \...
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1answer
55 views

Field of definition of a subvariety

Let $X$ be a variety over a field $\overline{k}$ (where $k$ is a field which is not algebraically closed). Each closed point of $X$ is defined over some finite extension $\ell/ k$, which means that it ...
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53 views

Non-smooth curve in $\mathbb{A}^2$

In one of my exercises on Algebraic Geometry, I showed that the curve $X \subset \mathbb{A}^2$ defined by $x^3-y^2$ is irreducible but not smooth. Furthermore, they ask the following question that I ...
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28 views

Show that $K(X)$ is not a field for $X = Z(xy)$

Let $X = Z(xy) \subset \mathbb{A}^2$. I want to show that the rational fuctions on $X$ defined as $ K(X) := \{(U,f) : U ⊂ X \text{ is open and dense and } f \in \mathcal{O}_X(U)\}/ \sim$, where $(U,...
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1answer
32 views

Characterization of linear normality

A projective variety $X\subseteq\mathbb{P}^{n}$ is said to be linearly normal if the restriction map $$ H^{0}(\mathbb{P}^{n}, \mathcal{O}_{\mathbb{P}^{n}}(1))\rightarrow H^{0}(X,\mathcal{O}_{X}(1)) $$...
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47 views

What is extension of scalars used for in algebraic geometry?

Given a ring homomorphism $f:A \rightarrow B$ and an $A$-module $M$, one can construct and $A$-module with the tensor product: $M_B=B \otimes_A M$ which has a $B$-module structure. This is said to be ...
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63 views

Take tautological bundle over $\mathbb{C}P^n$ less the zero section. Is this homeomorphic to $\mathbb{C}^{n+1}-{0}$?

It appears that (non zero vector in $\mathbb{C}^{n+1}$)$\mapsto$(line containing the vector, vector in this line) is a homeomorphism from $\mathbb{C}^{n+1}$ to the tautological bundle over $\mathbb{C}...
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31 views

Dominant morphism on affine varieties

Let $X,Y\in \mathbb{A}^{n}_{k}$ affine varieties, I know that a morphism $f:X\rightarrow Y$ is dominant iff the correspondent morphism $\phi:k[Y]\rightarrow k[Y]$ is injective. How can I show from ...
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2answers
63 views

Showing the ideal $\left \langle yz,xz,yx+ay,x^2+ax \right \rangle$ is radical for all $a\neq 0$

Let $I_a = \left \langle yz,xz,yx+ay,x^2+ax \right \rangle$ be an ideal of $k[x,y,z]$, where $a \neq 0$. Show that $I_a$ is radical. What is the geometric meaning of the elements in $\sqrt{I_0}\...
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1answer
51 views

All cohomology of quadrics comes from algebraic cycles

I've read in a number of place now the statement that all cohomology of quadrics (complex projective ones) comes from algebraic cycles, but I cannot find any source for this. So I hope someone here ...
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1answer
43 views

Zeros of specialization of a family of polynomials [closed]

Let $k$ be an algebraically closed field, and $K\supset k$ be an algebraically closed extension. Let $a\in K^n$ be a tuple, we call $a^\prime\in k^n$ a specialization of $a$ if for any $f(X)\in k[X]$ ...
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55 views

Definition of regular map from aquasi projective variety

These are from the book Basic Algebraic geometry by Shafarevich Definition of regular function on a quasi projective variety is as follows : Let $X\subset \mathbb{P}^n $ be a quasi projective ...
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28 views

A set of $d$ points in projective space may be described with polynomials of degree $d-1$.

This is Exercise 1.3 in Harris' Algebraic Geometry. Show that if $\Gamma \subset \mathbb{P}^n$ consists of $d$ points and is not contained in a line, then $\Gamma$ may be described as the zero ...
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21 views

The standard tropical lines - Pointwise valuation

Let $K$ be an algebraically closed field with valuation, and its value group $\Gamma_\text{val}^n$ is dense in $\mathbb{R}$. Consider the polynomial $ f(x,y)=x+y+1. $ It can easy by shown that the ...
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19 views

Integral basis of an extension of complete fields

Let $\mathcal{O}_K$ be a complete discrete valuation ring with quotient field $K = \text{Quot}(A)$. Let $L | K$ be an arbitrary finite field extension. Because $K$ is henselian, the integral closure $\...
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47 views

Why is the index $i(\mathcal{L})$ of an ample line bundle on an abelian variety equal to $0$?

I've seen that here https://www.math.uchicago.edu/~ngo/Shimura.pdf there's a theorem called Mumford's Vanishing Theorem (Theorem 2.2.2) which says: Let $\mathcal{L}$ be a line bundle on $X$ (...
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31 views

Parametric Equation of Elliptical Cycloidal Sine Curve

I am trying to find the parametric equations of a cycloidal curve, which, instead of using a circle, uses an ellipse to oscillate around a base circle. Below are equations of the standard, circular ...
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1answer
30 views

Define hyperplane out of 4 points

Given the Cartesian co-ordinates (x,y,z,w) of 4 non-coplanar points: $P1:(x1,y1,z1,w1)$ $P2:(x2,y2,z2,w2)$ $P3:(x3,y3,z3,w3)$ $P4:(x4,y4,z4,w4)$ I want to find the equation of the hyperplane on which ...
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Recommended books on commutative algebra stressing links with algebraic geometry

Can someone recommend some books on commutative algebra stressing links with algebraic geometry? My concern is this. It seems to me that most of commutative algebra was formulated at least initially ...
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1answer
41 views

Why is the polynomial $F \in K[Z_0, \dots, Z_n]$ on $K^{n+1}$ not a well defined function on $\mathbb{P}^n$ in general?

I'm reading Joe Harris' Algebraic Geometry and he says "A polynomial $F \in K[Z_0, \dots, Z_n]$ on the vector space $K^{n+1}$ does not define a function on $\mathbb{P}^n$" where $K$ is a ...
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24 views

Image of base change of immersion

I'm revising for my exams now and struggling with the following exercise: Let $f: X \to S$ be an open or closed immersion and $g: S' \to S$ another morphism where $X,S,S'$ are schemes. Then the base-...
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26 views

Genus of Curves over finte fields

This may be a dumb question but is calculating the genus of a curve define over a finite field different than over $\mathbb{C}$. For example the following curve: $y^8 + y +x^{12} + x^5$ is genus 14 ...
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35 views

How to prove that $i^\vee:X^\vee\rightarrow Y^\vee$ is dominant?

In my case $X$ is an abelian variety, $Y$ is an abelian subvariety of $X$, $i: Y\hookrightarrow X$ is the inclusion map,so we have $i^\vee:X^\vee\cong Pic^0X\rightarrow Y^\vee\cong Pic^0Y$, the dual ...
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1answer
45 views

The complete solution to a system of polynomials over $\mathbb{R}$

If I am solving a positive-dimensional system of polynomials over $\mathbb{R}$, and specifically am searching only for real solutions, how do I know that my solution is complete and there are no other ...
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33 views

How do you define the restriction of a sheaf?

Just to be clear with the notations: Recall that the pullback of $\mathcal{F}\in\mathcal{O}_B\text{-Mod}$ via $f:A\rightarrow B$ (morphism of schemes) is defined as \begin{equation*}f^*\mathcal{...
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48 views

Bounding the degree of an algebraic extension containing solutions to polynomials

Let $F$ be a field, and let $f_{1},\ldots, f_{s}$ be polynomials in $F[x_{1},\ldots, x_{t}]$. Assume that the degree of the polynomials is bounded by $d$, by which I mean, if $m$ is any term in $f_{i}$...
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39 views

Does an isogeny always define a covering map?

Consider a map $f: G_1 \to G_2$ between two topological groups. If $f$ is an isogeny when viewing $G_1,G_2$ as algebraic groups does $f$ always define a covering map when viewing $G_1,G_2$ as ...
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2answers
60 views

Help in showing that the cusp $(y^2-x^3)\subset \mathbb{C}^2$ is not isomorphic to $\mathbb{C}$

Let $X:=(y^2-x^3)\subset \mathbb{C}^2$ be the vanishing of the polynomial $f(x,y)=y^2-x^3.$ I have proved an exercise in Hartshorne: If $\varphi:\mathbb{C} \to X, \ t \mapsto (t^2,t^3)$ is the ...
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1answer
56 views

Understanding a proof of Hartshorne's book proposition 2.2.

I have been reading the book Algebraic Geometry by Robin Hartshorne and I have found the following proposition: For part b) the proof goes as follows: The thing is that, How can we ...
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1answer
47 views

Conics and conics of the form $ax^2+by^2+c=0$

The problem of finding rational points on conics is usually discussed (for example in the book of Silverman and Tate) for conics of the form $ax^2+by^2+c=0$. I assume that those conics are in ...