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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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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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 ...
3
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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
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 ...
13
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3answers
523 views

The algebraic de Rham complex

Let $A$ be a commutative $R$-algebra (or more generally a morphism of ringed spaces). Then there is an "algebraic de Rham complex" of $R$-linear maps $A=\Omega^0_{A/R} \xrightarrow{d^0} \Omega^1_{A/R} ...
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3answers
374 views

Geometrically, why do line bundles have inverses with respect to the tensor product?

Geometrically, why do line bundles have inverses with respect to the tensor product? Here my thoughts on the problem so far, please excuse their scatteredness. I know algebraically, it is just ...
5
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1answer
162 views

Weil does not imply Cartier on variety $X$.

Show that the divisor $D$ defined by $a = b = 0$ in the variety $X \subset \mathbb{A}^4$ defined by $ad - bc = 0$ $($the cone on a smooth quadric surface$)$ is not locally principal. My attempt ...
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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 ...
6
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1answer
74 views

Cuspidal cubic not coarse moduli space parametrizing one dimensional subspaces of $\mathbb{C}^2$?

Let $F$ be the functor of flat families of lines through the origin in $\mathbb{C}^2$. Let $C$ be the projective curve with plane model $y^2 = x^3$, i.e. in projective space it is defined by $y^2z = ...
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0answers
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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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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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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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 ...
3
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1answer
45 views

Prove that $IL,JK$ and angle bisector of angle $BCD$ are concurrent

Given a convex quadrilateral $ABCD$. In $\Delta ABC$, $I$ is the incentre and $J$ is the excentre opposite to vertex $A$. Similarly, $K$ is the incentre and $L$ is the excentre opposite to vertex $A$ ...
2
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0answers
68 views

Could any one explain the difference between the theorems?

In the paper http://annals.math.princeton.edu/2007/165-2/p04 Theorem 2. Let $b \ge 2$ be an integer. The b-ary expansion of any irrational algebraic number cannot be generated by a finite automaton. ...
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490 views
+50

Tate conjecture for Fermat varieties

I've been looking at Tate's Algebraic Cycles and Poles of Zeta Functions (hard to find online... Google books outline here) and have a question about his work on (conjecturing!) the Tate conjecture ...
5
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4answers
243 views

Are all groups algebraic?

I know the definition of a group as a set with an operation that satisfies certain axioms. I have heard that there is something called an algebraic group and that this is a group with a topology such ...
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0answers
36 views

Is the variety a closed subscheme of the fibered product?

Let $S$ be a surface over $\mathbb{C}$. And let $L$ be an ample line bundle on $S$. Let $C\in |L|$ be a smooth curve. And Let $A$ be a globally generated ample line bundle on $C$ with $n+1$ sections, ...
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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 ...
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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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2answers
115 views

Question regarding Vakil's algebraic geometry notes

Exercise 1.3 D of Vakil's lecture notes on algebraic geometry asks: "Verify that $A \to S^{−1}A$ satisfies the following universal property: $S^{−1}A$ is initial among $A$-algebras $B$ where every ...
3
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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.
4
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2answers
138 views

Open subset in Irreducible Topological Space is dense.

Show that every non-empty open subset of an irreducible topological space is dense. I know a lemma that states that $U \subset$X is dense iff for all $A \in \tau$, $A \cap U \neq \emptyset$. So ...
1
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1answer
347 views

The cone over a projective variety

I'm trying to prove that $I(C(Y))=I(Y)$, where $C(Y)=\pi^{-1}(Y)\cup \{(0,\ldots,0)\}$ the cone over $Y$ and $\pi:\mathbb A^{n+1}-\{0,\ldots,0\}\to \mathbb P^n$ the projection which sends the point ...
14
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2answers
324 views

Which continuous functions are polynomials?

Suppose $f \in C(\mathbb{R}^n)$, the space of continuous $\mathbb{R}$-valued functions on $\mathbb{R}^n$. Are there conditions on $f$ that guarantee it is the pullback of a polynomial under some ...
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1answer
32 views

question about the dimension of the global section space of a vector bundle

Suppose that $L,L^{'}$ are a line bundle over a compact riemann surface $C$. Take $H^0(C,L\otimes L^{'})$. Is it true that $h^0(C,L\otimes L^{'})=h^0(C,L)+h^0(C,L^{'})$ where $h^0(V)$ ,means the ...
3
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0answers
52 views

The form of the zeta function of an elliptic curve over a finite field

I seek a (very) elementary proof that the zeta function of an elliptic curve $E$ over $\mathbb{F}_q$ has the form $$Z(T)=\frac{1-aT+qT^2}{(1-T)(1-qT)}.$$ Something tedious and computational making use ...
21
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4answers
2k views

Intuitive explantions for the concepts of divisor and genus

When trying to explain AG-codes to computer scientists, the major points of contention I am faced with are the concepts of divisors, Riemann-Roch space and the genus of a function field. Are there any ...
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3answers
1k views

Toy sheaf cohomology computation

I asked this question a while back on MO : One thing that really helped in learning the Serre SS was doing particular computations (like $H^*(CP^{\infty})$) I am curious, as a sort of followup if ...
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0answers
45 views

Vector bundles on Hirzebruch surface $\mathbb{F}_2$

I would like to know a classification for all holomorphic vector bundles on the second Hirzebruch surface $\mathbb{F}_2$. Is this known? What is known? In particular, I'm looking for holomorphic ...
1
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1answer
33 views

References about moduli space of abelian varieties with level structure

In the course of one of my research project, I have been advised to try to have a look to "Moduli Space of Abelian Varieties with Level Structure". I am interested in references where this topic is ...
2
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0answers
45 views

Constructing $\text{Hilb}_P(X/S)$ as a locally closed subscheme of $\text{Hilb}_P(\mathbb{P}^n/S)$?

For a projective scheme $X/S$, how do I construct $\text{Hilb}_P(X/S)$ as a locally closed subscheme of $\text{Hilb}_P(\mathbb{P}^n/S)$? ($P$ is Hilbert polynomial.) Can I get a reference to this ...
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0answers
23 views

interpreting certain coefficients as coordinates

By means of Pluecker coordinates, there is a $1-1$ mapping between all lines of $\mathbb{P}^3$ and a certain quadratic hypersurface $\Pi$ of $\mathbb{P}^5$. Let $X$ be a curve of $\mathbb{P}^3$ and ...
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1answer
27 views

Weighted sum of angles modulo $\pi/2$

Angle modulo $\pi /2$ means: $(a+ \pi /2) \mathbin{\%} \pi/2=a$, $a \in [0, \pi/2)$, which could be illustrated as a ‘modulo circle’ in the following figure. How to calculate the weighted sum of a ...
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0answers
17 views

Algebraic conditions for the directions coming from a hyperbolic configuration of point

Consider hyperbolic $3$-space $H^3$, thought of as the open unit ball in $\mathbb{R}^3$, where geodesics are represented by arcs of circles etc. (the well known Poincare model of $H^3$). Let $B$ ...
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1answer
64 views

Maximal ideals and the projective Nullstellensatz

This is a simple question, but it's one of those things that I've been thinking about so much that I've just kind of lost where I am and need some explicit reference. One of the main corollaries of ...
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0answers
25 views

Pole of a tangent

Theres an algebraic curve and a tangent line through the point $P$ of that curve. I have been trying to find a pole of a tangent line but I couldnt manage. Any ideas?
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0answers
38 views

Mimimal equations defining a linear subspace (or: how I forgot linear algebra).

i have a question that may be trivial, but I just can't find the answer in the internet (nor in my head). Given a linear vector subspace of fimension $d$ and given his Plücker coordinates i can ...
4
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1answer
77 views

Related to Hartshorne Exercise 2.4.3, nothing to do with separatedness or properness.

Let $U = \text{Spec}\,A$ and $V = \text{Spec}\,B$ be open affines in a scheme $X$ (not necessarily separated). How do I show that for each $P \in U \cap V$ there is an open affine $W$ such that $P \in ...
2
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1answer
56 views

Help with my proof of maximum number of intersection points of n lines being n(n-1)/2

I know that this question is a hoary old chestnut, but I have never seen a proof before working one out myself, so I'd like you to help me see if mine is rigorous enough. Obviously with $n$ linear ...
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0answers
43 views

Question about points of a variety lying in an extension as K-morphisms

I hope that someone can shed some light on this for me.. or at least point me to some references. Suppose that $X$ is an algebraic (let's just say affine) variety defined over $k$. Suppose I have a ...
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21 views

Help to understand the proof of the Riemann Munford relation

Here i post a file where from page 617 to 618 there is the proof of the Riemann mumford relation that is the theorem 1.13. My problem is to understand the beginning of that proof. In particular ...
2
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1answer
44 views

Real Points of resolution of singularities of $\mathrm{Spec} \mathbb{R}[x,y]/(x^2+y^2)$

Consider the scheme $X = \mathrm{Spec} \mathbb{R}[x,y]/(x^2+y^2)$. Scheme-theoretically, it's a one dimensional scheme with one real, singular point (there are, of course, other complex points). I'm ...
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80 views
2
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0answers
55 views

What is the type of the surfaces $x^5 - y^5 + z^2 + x=0$ and $x^5 - y^5 + z^2 + x+1=0$?

I am interested what is the type of the surfaces over the rationals $$ x^5 - y^5 + z^2 + x=0$$ and $$ x^5 - y^5 + z^2 + x+1=0$$ Magma's ...
4
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1answer
330 views

Lines in $\mathbb{A}^3$

This seems intuitive, but I'm having trouble coming up with an exact matrix for the problem. Let $\{L_1, \ldots, L_N\}$ be a set of lines through the origin $(0,0,0)$ in the affine space ...
2
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0answers
50 views

$V$-bundles and vector bundles

I am looking for more information on $V$-bundles. They are hard to search for as either vector bundles come up or something like GL($V$)-bundles come up. I am looking for some nice expository ...
1
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1answer
76 views

$\mathbb{A}^2\setminus (0,0)$ is not affine

I want to prove that $X = \mathbb{A}^2\setminus (0,0)$ is not affine. My attempt: If $\Bbbk[X] = \Bbbk[x,y]$ then $X$ is not affine since $(x,y) \subset \Bbbk[x,y]$ is a proper ideal, but $V(x,y) ...
5
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1answer
69 views

References for the threefold categorical equivalence of compact Riemann surfaces?

A lot of the books I've found assert that there is a threefold categorical equivalence between (1) compact Riemann surfaces, (2) smooth projective algebraic curves, and (3) function fields of ...
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26 views

MCM Modules over Cyclic Quotient Singularities

Let $k$ be a field and $R$ the ring $k[[u^{n+1}, uv, v^{n+1}]]$. Then the indecomposable MCM $R$-modules are given by $M_j = R(u^av^b \vert b-a\equiv j \mod{n+1})$ for $j = 1,\ldots, n$. This is of ...