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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26 views

Barring a morphism to subvarieties

This is exercise I.3.10 from Hartshorne.I understand that restrict a morphism is continuous but not understand the topological structure of a locally closed irreducible in connection with regular ...
2
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1answer
47 views

Proof of Chow's lemma in EGAII

Section 5.6 of EGAII is dedicated to Chow's lemma. I am having a hard time following an early step of the proof. The version of Chow's lemma in the text assumes that $X$ is a separated $S$-scheme of ...
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2answers
25 views

Dickson's Lemma (proof of Prop. 2.23 in Hasset's Intro to Alg Geom)

I'm studying Hasset's book by myself but I had no previous formal algebra training. To prove Dickson's lemma (prop. 2.23, p. 19) he defines the auxiliary monomial ideals $$J_m=\left<x^\alpha \in ...
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0answers
19 views

How to find location - multilateration

I have this data: $$ {x1} = 473463,100288[m]\\ {y1} = 5924242,046998[m]\\ {z1} = 0[m]\\ {t1} = 41919,84025[s]\\ {x2} = 473483,237020[m]\\ {y2} = 5924212,730018[m]\\ {z2} = 0[m]\\ {t2} = ...
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0answers
31 views

Line bundles on $\mathbb{P}^2 \times \mathbb{P}^2$

I'm curious as to what are all the line bundles on $\mathbb{P}^2 \times \mathbb{P}^2$? I might be mistaken but I believe they are classified as $$ \mathcal{O}_{\mathbb{P}^2 \times \mathbb{P}^2}(p,q) ...
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0answers
20 views

Abelian hypersurfaces

Is there any way to tell from its defining equation when an affine hypersurface has a group law? I am aware that all such groups are matrix groups; in that case I might want to ask when a variety over ...
3
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28 views

The symmetric product of elliptic curves

Suppose $E$ is an elliptic curve, what is the symmetric product $F=E\times E/S^2$? It is a smooth surface, let $\pi\colon E\times E\to F$ be the projection, then we have ...
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0answers
29 views

Automorphisms of cubic nodal curve

How to calculate the automorphism group of the nodal cubic curve $y^2=x^3+x^2$ ? Should I use the rationality of this cubic curve ?
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32 views

When is “number of points in the fiber” semicontinuous?

Let $f:X \rightarrow Y$ be a finite morphism of schemes. For $y \in Y$ let $n(y)$ be the cardinality of the set-theoretic fiber $f^-1(y)$. This is finite since $f$ is finite. I think that the ...
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0answers
20 views

Global section on product of sheafifications

Say I have two presheaves $\mathcal F$ and $\mathcal G$ of $\mathcal O_X$-modules on a scheme $X$. Let $\mathcal F^+\!, \mathcal G^+\!$ be their respective sheafifications. We then have a commitative ...
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1answer
42 views

Is the quotient morphism from product of curves to to their symmetric product flat?

Suppose $C$ is a smooth curve, is the morphism $C^2=C\times C\to C^{(2)}=C\times C/S_2$ flat? What about the general case?
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34 views

Blowing up fibers in families - looking for comparison results

Given a morphism $\pi: S\rightarrow B $, an ideal sheaf $I$ on $S$, and a point $b\in B$, I wish to consider the blow up of $S$ along $I$. Say I know something about the pullback $I(b)$ to the fiber ...
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0answers
30 views

Simple question about closed immersions

I would like to know if I understand closed immersions correctly. Let $f:X \rightarrow Y$ be a morphism of schemes (or locally ringed spaces in general). The morphism $f^\sharp:\mathscr{O}_Y ...
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1answer
54 views

Dimension of $\mathfrak{m}^k/\mathfrak{m}^{k+1}$?

Let $C \subset \mathbb{P}_\mathbb{C}^2$ be a projective curve, and $p \in C$ a point of multiplicity $m$. If $\mathcal{O}_p(C)$ is the local ring of $C$ at $p$, and $\mathfrak{m} \subset ...
3
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1answer
77 views

Galois invariants of the Tate module of an elliptic curve over a number field

Let $K$ be a number field, $E$ be an elliptic curve over $K$, $l \neq p$ be two different prime numbers and $v$ be a place of $K$ above $l$. I am trying to understand the proof of proposition I.6.7 ...
3
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1answer
24 views

Nonsingular cubic curve, quotient of $d(x/z)$ and $y/z$ is differential which is regular everywhere.

Let $C \subset \mathbb{P}_2$ be the cubic curve defined by$$y^2z = x(x-z)(x-\lambda z)$$with $\lambda \in \mathbb{C} - \{0,1\}$. Let $p = [0, 0, 1]$, $q = [1, 0, 1]$, $r = [\lambda, 0, 1]$, and $s = ...
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28 views

Normal bundle of zero scheme of section

Suppose $Y$ is a smooth variety, $E$ is a rank $d$ bundle on $Y$, $s$ is a regular section of $E$ over $Y$,(i.e.,locally under a trivialization $E|_U\cong O_U^d$, write $s=(s_1,\dots,s_d)$, then $s_i$ ...
2
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0answers
41 views

Ideals in a ring as geometric objects?

I am interested in learing about the possibility of (one-sided) ideals in a ring being repreented geometrically. In other words, about their status as geometric objects (after all, they can be dealt ...
3
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29 views

About an example of normal bundle of a curve over a surface

I know the definition of the normal bundle $N_{C/S}$ of a curve $C$ over a surface $S$ as the cokernel of the injection $T_C \subset T_S|_C$ where $T$ is the tangent bundle. I would like to exhibit an ...
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0answers
16 views

Compactifying affine algebraic families

Suppose I have a smooth morphism $f:X\to S$ of affine varieties over an algebraically closed field of arbitrary characteristic. I want to regard this as a family of varieties parametrized by $S$ and ...
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1answer
40 views

Proof that $H^{0,1} \oplus H^{1,0} = H_{DR}^1$

I am struggling with a proof from Donaldson's Riemann Surfaces which he leaves as an exercise. we want to construct an isomorphism from the direct sum of $H^{1,0}(X)$, the set of holomorphic 1-forms ...
2
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2answers
67 views

Sum of Betti numbers of a degree $d$ hypersurface of $\mathbb{P}^n_{\mathbb{C}}$

Let $X \subseteq \mathbb{P}^n_{\mathbb{C}}$ be the zero set of a homogeneous polynomial of degree $d$. Assume that $X$ is smooth. Is there a way to find the sum of the Betti numbers of $X$ (i.e. the ...
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1answer
39 views

Riemann-Roch analysis of point divisor ring on smooth genus 3 Riemann surface

Let $C=C_4\subset\mathbb{P}^2$ be the smooth genus 3 Riemann surface given by a quartic curve. Let $P\in C$ be a point, and $D=P$ the divisor given by the point $P$. Let ...
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0answers
7 views

How can a nonconvex polytope be defined (not by an LMI)?

A convex polytope can be defined by an LMI (linear matrix inequality) or a list of points. How can a nonconvex polytope be defined?
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27 views

Maximum volume inscribed ellipsoid inside nonconvex polyhedron

In Convex Optimization (Boyd, Vandenberghe), an algorithm for finding the maximum volume inscribed ellipsoid inside a convex polyhedron is given on p. 414 (8.4.2 Maximum volume inscribed ellipsoid) [I ...
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35 views

Is there affine surface of general type of the form $y^2=f(x) f(z)$ or $y^2=f(x) g(z)$?

Let $f,g$ be univariate polynomials with integer coefficients of degree $n$. Is there affine surface of general type of the form (1) $y^2=f(x) f(z)$ or (2) $y^2=f(x) g(z)$? I would expect for $n$ ...
2
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1answer
46 views

Questions about algebraic curve definition

The algebraic curve definition states as following $S_{2}^m$ denotes homogeneous polynomial of degree $m$ in $x$ and $y$ $f_{m}(x, y) = \sum_{j, k \ge 0, j+k=m} C_{j,k}x^{j}y^{k}$ The elements of ...
2
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1answer
63 views

Every commutative ring is a quotient of a normal ring?

In the book Étale cohomology by Milne I found on p. 37 (in the context of constructing the henselization of a local ring) the following claim: "Every ring is a quotient of a normal ring". The same is ...
2
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1answer
36 views

What is the intersection of the Segre variety in $\mathbb{P}^5$ and the Veronese surface in $\mathbb{P}^5$?

This is an exercise from Chapter 8 of Ideals, Varieties and Algorithms by Cox et al. The projective Veronese surface in $\mathbb{P}^5$ is defined as the projective closure of the surface $S$ which ...
2
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0answers
50 views

Proofs of Hodge duality: $H^{0,1}(X) = H^{1,0}(X)^*$

I am looking for a proof of this fact, where $H^{1,0} = Ker(d: \mathscr{E}^{1,0} \rightarrow \mathscr{E}^{2})$ and $H^{0,1} = Coker(\overline{\partial}: \mathscr{E} \rightarrow \mathscr{E}^{0,1}$, ...
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1answer
70 views

Complementary textbook algebraic geometry

I don't know where to ask this or if it is allowed to do it, so please let me know any details for further questions of this kind. I am taking an algebraic geometry class and am using the textbook ...
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38 views

Complexification of proper scheme

Let $X$ be a proper scheme over $\mathbb{C}$. We define $X_{\mathbb{R}}$ to be a scheme over $\mathbb{R}$, which is the same topological space as $X$ with structure sheaf generated by real and ...
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2answers
50 views

Finding parabola parameter given 2 points

How can I determine which is the directrix and the focus of a parabola and what is the distance between those points, only knowing that this parabola has its symmetry axis = OX and its passes through ...
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0answers
43 views

Proof of Serre duality for $D=0$

I have been working through a proof of Serre duality, which proceeds by induction on the divisor $D$, but I am having trouble with the base-case. How can I prove that on a riemann surface X, $H^0(X, ...
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0answers
33 views

Dimension zero of Cohomology Group

Let $X$ be a complete non-singular curve and $\Omega^1$ the sheaf of regular 1-forms on $X$. Let $g=\dim H^0(X, \Omega^1)$ the genus of $X$. I've read that there are $g$ points $P_1,\cdots, P_g$ on ...
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0answers
36 views

About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$

I've been trying to find some useful categorical facts about the category of schemes, locally ringed spaces and ringed spaces (that I shall denote by $\text{Sch}$, $\text{LRS}$ and $\text{RS}$ ...
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0answers
30 views

Subsheaves of locally free sheaves on a rational curve

Let $k$ be an algebraically closed field, $\mathcal{E}$ a locally free sheaf on $\mathbb{P}^1_k$ and $\mathcal{L} \subset \mathcal{E}$ a subsheaf of rank $1$. By Grothendieck's theorem, we know that ...
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2answers
43 views

Application of Hilberts Nullstellensatz (strong form)

Let $\langle f_1,\ldots,f_r \rangle $ be an ideal in $\mathbb{C}[x_1,\ldots,x_n]$. Then an element $g \in \mathbb{C}[x_1,\ldots,x_n]$ belongs to $\sqrt{\langle f_1,\ldots,f_r \rangle}$ if and only if ...
2
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0answers
36 views

Global sections of sheafification of Cohen-Macaulay module

Let $S=k[x_0,\ldots,x_n]$ be the polynomial ring over a field $k$ with the standard grading. Let $M$ be a finitely generated graded Cohen-Macaulay $S$-module of dimension at least two. Let ...
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10 views

Closure of Schubert cell is the Schubert variety

My question concerns Proposition 1.4.6 in the following article: http://www.mi.uni-koeln.de/~littelma/SMTkurz.pdf . There's just one, apparently straightforward detail of the argument which I can't ...
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76 views

May Algebraic Geometry be appropriate for me? [closed]

I am a student of Mathematics who have to choose its area of specialization. I am trying to obtain as more information as possible, by asking a lot of questions to more experienced people, trying to ...
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0answers
47 views

Two simple counterexamples in algebraic geometry

Suppose we have a smooth complex algebraic variety $X$. Then in general, $K^a(X)\to K(X^{an})$ is not surjective. Could someone give an example of a topological vector bundle class which contains no ...
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63 views
+50

Siegel's Theorem for Genus 0 Curves over Function Fields

Let $V$ be a quasi-projective algebraic curve over a field $k$. Are there finitely many morphisms from $S$ to the triply-punctured sphere $\mathbb{P}^1 -\{0,1,\infty\}$? This is true if $k = ...
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0answers
21 views

Reference for a couple of terms, $\underline{\operatorname{Hom}}_X(-,-)$ and $\boxtimes$

I have a couple of questions on symbols. What are the names for $\underline{\operatorname{Hom}}_X( \mathscr{F},\mathscr{G})$ for sheaves on a scheme $X$, and $\boxtimes$? And what would be a ...
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1answer
56 views

Spectrum of the Ring of Continuous Functions on a Space

I was wondering when exactly we can recover the topological space, $X$, from its ring of continuous functions into $\mathbb C$ (or some sort of sufficient topological group). For any topological ...
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1answer
62 views

Dimension of $\mathbb{Q}$-vector spaces $H^m(X, \mathbb{Q})$.

Assume that you can't compute the cohomology group $H^m(X, \mathbb{Q})$ for$$X = \{(x : y : z : w) \in P^3(\mathbb{C}): xy = zw\}$$but you know Weil conjecture. By using Weil conjecture, give the ...
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1answer
46 views

Why is this a tori

In her notes http://www.math.toronto.edu/fiona/courses/algp.pdf on page 383, Example 4.2 Fiona claims that the group $$ T = \left\lbrace \pmatrix{ a & b \\ -b & a } \bigg|\, a,b \in ...
3
votes
1answer
50 views

Example of a homeomorphic regular morphism of affine algebraic sets that's not an isomorphism of affine algebraic sets?

As the title suggests, can anyone give me an example of a homeomorphic regular morphism of affine algebraic sets that is not an isomorphism of affine algebraic sets? Many thanks in advance.
1
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1answer
52 views

'Proof' of the correspondence between maximal ideals and points in projective space

The affine Nullstellensatz tells us that we have an inclusion-reversing bijection between radical ideals of $A=k[x_1,\ldots,x_n]$ and affine varieties of $\mathbb{A}^n$, given by $\mathbb{V}\colon ...
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0answers
3 views

Hermitian Matrices over Quaternions with Rank at most k

The set of Hermitian matrices of the form: $X+iY+jW+kZ$ with $X,Y,Z,W \in \mathbb{C}^{M x M}$. $X$ symmetric, and $Y,Z,W$ skew-symmetric, with $rank(X+iY+jW+kZ)\leq{k}$, has what dimension as a ...