The study of geometric objects defined by polynomial equations. Algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc.
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1answer
77 views
Sheaf of Ramification Divisor - Hurwitz Formula
This question refers to the proof of Proposition 2.3/IV in Hartshorne, page 301.
Let $R$ be the ramification divisor associated to a finite, separable morphism of curves $f:X \rightarrow Y$. Why is it ...
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1answer
41 views
What is a generic hyperplane of a projective space over $\mathbb{C}$?
I hear sometimes about a generic hyperplane of a projective space over $\mathbb{C}$, for example in the Noether-Lefschetz theorem.
What is the definition of it?
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1answer
39 views
Projective Equivalence of $(n+2)$-tuples in $\mathbb{P}^n$
Let $p_1,...,p_{n+2}$ and $q_1,...,q_{n+2}$ be two sets of distinct points in general position in $\Bbb P^n$ (there may however be overlaps between the two sets.). Then there exists $\phi\in ...
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1answer
25 views
Finding generators for an ideal consisting of the set of functions vanishing on a subset of Spec
Let $U$ be the union of the $x,y,z$ axes in complex affine 3-space. The set of functions that vanish on $U$ is an ideal. Can we neatly express their generators?
There are a lot of similar such ...
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1answer
118 views
Decomposition of Noetherian space into irreducible subsets
I am trying to relate two (maybe not) different decompositions of a noetherian topological space into irreducible subsets, given in Ravi Vakil's notes on algebraic geometry.
Exercise 4.6.N : Let ...
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1answer
70 views
For open affine $U\subset$ affine $X$, why is $k[U]$ flat over $k[X]$?
This is a homework problem for me so please do not post a full solution. I would very much appreciate a hint to move me past the point at which I'm stuck. Here is my work so far:
The setting is that ...
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1answer
51 views
How to recognize a glueing
This is an exercise in chapter 1 of Fulton's book "Introduction to toric varieties".
Let $\Delta$ be the fan consisting of the cones $\sigma_1=\langle e_1, e_2\rangle$ and $\sigma_2=\langle ...
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1answer
59 views
$\mathcal{O}_{X}(d)\simeq \mathcal{O}_{X}(D)$?
On $\mathbb{P}^n$ let $D$ be a smooth hypersurface defined by the equation $F=0$, F an homogeneous polynomial.
$\mathcal{O}_{\mathbb{P}^n}(D)$ is the sheaf of meromorphic functions on $\mathbb{P}^n$ ...
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1answer
46 views
Counting hypersurfaces in P^n containing a set of points
The set of cubic hypersurfaces in $\mathbb{P}^3$ are paramatrized by a 19-dimensional projective space. If we consider only the hypersurfaces containing a fixed generic set of 19 points, it seems to ...
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2answers
130 views
Sheafs and closed immersion
Let $f:X \rightarrow Y$ be a continuous map of topological spaces, such that it is closed immersion. Let $\mathfrak{F}$ and $\mathfrak{G}$ be sheafs on $X$ and $Y$ respectively.
How to show, that ...
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1answer
41 views
One construction about sheafs
Let $(X,O_X)$ be a ringed space, $E$ - finite locally free $O_X$-module.
Let $E^*=Hom_{O_X}(E, O_X)$. How to show, that $E^{**} = E$?
It's clear, that locally $E|_U = O_X^n|_U$, and then $E^*|_U = ...
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2answers
66 views
Functional sheaf (Hartshorne, Cartier divisors)
In Hartshorne there is the following description of the sheaf $K$ on the scheme.
For each open $U = Spec \, A$ we define $K(U) = S^{-1} A$, where $S$ is the set of non-zero-divisors. Why is it a ...
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1answer
86 views
Deformation to the normal cone
Fulton in his book "Intersection theory" uses local description of this deformation that I can't understand. I quote paragraph from page 87 and insert my questions.
Assume $Y=\operatorname{Spec}(A)$, ...
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1answer
147 views
Krull dimension and transcendence degree
What is the simpliest proof of the fact,
that integral algebra $R$ over a field $k$
has the same Krull dimension as transcendence
degree $deg.tr_k R$?
Is it possibple to use only Noether normalization ...
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1answer
64 views
Are smooth relative curves over an arbitrary base normal?
Let $X/S$ be a smooth, projective scheme of relative dimension one over a scheme $S$ (which we may assume is affine Noetherian, but need not be reduced nor irreducible nor even connected). For $s \in ...
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1answer
144 views
Explaining projective space to master students
I am teaching an introductory course in algebraic geometry for masters and it turns out that many of them are not at all familiar with the notion of projective space. So it is necessary to spend ...
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1answer
43 views
Definition of homomorphism between divisor groups of curves
This question refers to the Definition in p. 137 from Hartshorne.
Let $f: X \rightarrow Y$ be a finite morphism of nonsingular curves.
Let $Q$ be a closed point of $Y$ and $P \in X$ such that ...
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1answer
52 views
Dimension of local rings at closed points in an irreducible scheme
Let $X$ be an irreducible separated scheme of finite type over an algebraically closed field $k$.
In the proof of Theorem 8.15 Hartshorne claims that for any closed point $x\in X$ we have ...
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1answer
66 views
Relation of Function Field of a scheme to the Local Ring of its Prime Divisor
Refer to p. 130 in Hartshorne: Let $X$ be a noetherian, integral separated scheme, regular in codimension 1, and let $Y$ be a prime divisor of $X$, with generic point $\eta$. Let $\xi$ be the generic ...
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1answer
52 views
Singular affine real varieties are no manifolds?
The curve $C \colon x^3 + x^2 = y^2$ is a singular affine variety with a node at zero. How would one show that as an real affine variety $C \subseteq \mathbb{A}_\mathbb{R}^2 = \mathbb{R}^2$ it is no ...
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1answer
61 views
A cohomology group- What does the $F^2$ denote?
I ran across another denotion I am not familiar with while studying Bloch's conjecture, with the 2nd cohomology group being given as $w'\in F^2H^2(X,\mathbb{C})$. I understand what the part on and ...
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1answer
61 views
Kaehler-Einstein metric on Calabi-Yau manifold
I am reading "Complex geometry" by D. Huybrecht. On p.223 the books says that "If $c_{1}(X)=0$, e.g. if the canonical bundle $K_{X}$ is trivial, and $g$ is Kaehler-Einstein metric, then Ric$(X,g)=0$. ...
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1answer
127 views
Ramification of covering maps of curves and the étale fundamental group
I'm interested in finite galois covers $\varphi: Y \rightarrow X$ between smooth proper curves over an algebraically closed field of characteristic zero, which are étale outside a prescribed finite ...
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1answer
63 views
Detecting the type of singularity with the Jacobian
Say we have a plane curve $\mathcal{C} = V(f(x,y)) \subset \mathbb{A}^2_{\mathbb{C}}$. The partial derivatives tell us about the singularities: if they all vanish at a point $p \in\mathcal{C}$ then ...
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1answer
75 views
Degree of varieties in $\mathbb{P}^n$
The degree of a variety $X$ of dimension $r$ is defined by $r!$ times the leading coefficient of its Hilbert polynomial. This is the defination given in Hartshorne, but I find it is very hard to ...
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2answers
111 views
Principal ideals having embedded components
Does there exist a noetherian domain $A$ and a principal ideal $I = (x)$ in it having an embedded component?
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1answer
118 views
Showing holomorphic functions are preserved under pullback by a holomorphic map
Let $f: X\rightarrow Y$ be a holomorphic mapping of complex manifolds and assume for simplicity that $dim(X)=dim(Y)=1$.
I want to show that it preserves holomorphic functions under pullback.
We define ...
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1answer
61 views
Relation between DVR's of a local domain and localizations of its integral closure.
$\textbf{1.}\,\,\,\,\,\,\,\,$
Let $(A,\mathfrak m_A)$ be a one dimensional local domain and let $B$ be its integral closure in the fraction field $L=\textrm{Frac}\,A$. Assume that $B$ is finitely ...
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1answer
101 views
Projective dimension of tensor product $M\otimes M$
If the projective dimension of an $R$-module $M$ is finite, then can we say that projective dimension of tensor product $M\otimes M$ (as an $R\otimes R$-module) is finite?
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2answers
130 views
Torsor whose ring of function is a field
Let $G$ be an affine group scheme over $\mathbb{Q}$. Then it is easy to see that if the ring of regular functions $H^0(G,\mathcal{O}_G)$ is a field then $G$ is the trivial group.
Let $P$ be a ...
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1answer
47 views
Powers of Zariski-dense subset span symmetric power
I wonder if something like the following is true: let $V$ be a finite-dimensional vector space over a field of characteristic zero and $S \subset V$ a Zariski-dense subset. Does $\{ v^d \ | \ v \in S ...
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1answer
47 views
Codimension of the complement of a quasi-affine open subset of a variety
It is a known fact from Algebraic Geometry that the complement of an affine open subset of a variety is of pure codimension one. Does the same hold for the complement of a quasi-affine open subset of ...
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1answer
81 views
Degree of Hessian surface invariant under linear transformations?
Given a surface $V(f) \subset \mathbb{P}^n$ for a homogeneous polynomial $f$ of degree $d$ on $\mathbb{P}^n$ and a linear transformation $g \in SL(n+1)$. Is the degree of the Hessian $H_f = V(\det ...
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1answer
126 views
relating flatness, equidimensional, and complete intersection
I am a bit confused and am trying to clarify some notions. First consider the following well-known statement.
A dominant map $f:X\rightarrow Y$ between regular varieties is flat if and only if it is ...
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1answer
89 views
smooth quotient
Let $X$ be a smooth projective curve over the field of complex numbers. Assume it comes with an action of $\mu_3$. Could someone explain to me why is the quotient $X/\mu_3$ a smooth curve of genus ...
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1answer
290 views
Saturated ideal
Let $k$ be a field, let $I \triangleleft k[X_1,\dots,X_n]=S$ be an ideal and fix $f \in S$.
The saturated ideal of $I$ is $I^{sat}=I:f^\infty=\{g \in S \mid \exists m \in \mathbb{N} \ s.t. \ f^mg \in ...
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1answer
78 views
Silverman's Lefschetz Principle
Let K be a field of characteristic 0, E/K an elliptic curve. The "Lefschetz principle" implies that $E[m] \simeq \mathbb{Z}/m \times \mathbb{Z}/m$, but for this to follow from the result for complex ...
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1answer
84 views
divisors on relative projective space
Let $X$ be a separated irreducible variety which is regular in codimension $1$ (I want to talk about Weil divisors), and write $\mathbb{P}^n_X = \mathbb{P}^n_k \times X$ for the projective space over ...
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1answer
71 views
regular functions: two definitons
Let $X$ be an (affine) algebraic set i.e. the zeros' locus of a set of polynomial $S\subseteq k[X_1,\ldots,X_n],$
Let's look at these two definitions:
1) A regular function in $p\in X$, is an ...
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1answer
131 views
Birational map from a variety to projective line
This is exercise $4.4$ part (c) of Hartshorne's book. Let $Y$ be the nodal cubic curve $y^{2}z=x^{2}(x+z)$ in $\mathbb{P}^{2}$. Show that the projection $f$ from the point $(0,0,1)$ to the line $z=0$ ...
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1answer
77 views
Finite Fibers over closed points
Let $f:X \rightarrow Y$ be a morphism of algebraic varieties over an algebraically closed field. If all fibers $f^{-1}(y)$ with $y$ closed point in $Y$ are finite, can one conclude that an arbitrary ...
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1answer
80 views
to show it is Cohen -Macaulay Ring
Could anyone give me hint How to show this one: Let $V$ be a finite set of points in projective space.How to show that the coordinate ring of $V$ is Cohen Macaulay?
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1answer
187 views
Intersection number of plane curves - Motivation
I was reading the wikipedia page on intersection number: http://en.wikipedia.org/wiki/Intersection_number#Intersection_multiplicities_for_plane_curves
and I was confused by property number 6 they ...
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1answer
65 views
tangent space at origin of a variety
Could any one explain me how to show that the tangent space at origin of the variety $V=\mathbb{V}(y^2-x^3)$ is equal to full affine plane? They have defined $l$ is a tangent line at $p$ if the ...
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1answer
51 views
Understanding a morphism of modules by properties of the induced residue field homomorphism
Let $A$ be a reduced local Noetherian ring, and $\phi: M\to N$ a morphism of finitely generated free $A$-modules. For all $\mathfrak{p}\in\text{Spec}(A)$, let ...
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1answer
86 views
How to find the canonical divisor on a nonsingular toric variety?
I am reading Fulton's "Toric Varieties." In it, he explains that if $X$ is a toric variety and if $D_1, \ldots, D_d$ are the irreducible divisors invariant under the big torus action, then
$$ ...
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2answers
67 views
If $Y = X \backslash Z(f)$ for some affine variety $X$ with $p \in Y$, then $T_p Y \cong T_p X$
Let $Y = X \backslash Z(f)$ be a quasi-affine variety for some affine variety $X$, and let $p \in Y$. I'd like to prove that $T_p Y \cong T_p X$.
I have the following definition of $T_p X$:
If ...
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1answer
117 views
Is there a fundamental domain for $\Gamma(2)$ contained in the following strip
Let $\Gamma(2)$ be the subgroup of $\mathrm{SL}_2(\mathbf{Z})$ satisfying the usual congruence conditions. It acts on the complex upper half-plane.
Does it have a fundamental domain contained in the ...
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1answer
55 views
Explicit polynomials for a subvariety of $X\times \textbf P^{n\ast}$.
We work over an algebraically closed field $k$. Let $X=V_+(f_1,\dots,f_r)$ be a (smooth, if you want) projective subvariety of $\textbf P^n$, so $f_i(x_0,\dots,x_n)$ are homogeneous polynomials. I ...
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1answer
135 views
An irreducible polynomial over $\mathbb{R}[x,y]$
Let $(x_i,y_i)$ be a finite subset of points of $\mathbb{R}^2$. Find an irreducible polynomials $f(x,y)$ over $\mathbb{R}[x,y]$ such that vanish only in that points.
EDITED: Where $\mathbb{R}$ ...
