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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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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+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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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
57 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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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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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
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.
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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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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 ...
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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 $A$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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46 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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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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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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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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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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65 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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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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39 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 ...
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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 ...
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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 ...
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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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80 views

Is this surface rational — Magma says no, but I have rational parametrization?

Let $f(x,y,z)$ be the degree $6$ polynomial: ...
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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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$V$-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 ...
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1answer
37 views

Translation of 'morphisme net'?

In French, one refers to a certain 'morphisme net'. I am looking for the English translation of this. EDIT: The term appears here on p.22 Lemme 2.7.2. Unfortunately I have not been able to find the ...
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158 views

Weighted projective space and $\mathrm{Proj}$

I'm trying to solve a problem in Jenia Tevelev's notes on GIT. (It can be found as Problem 5 at the end of this pdf.) Compute $$\operatorname{Proj}\frac{\mathbb{C}[x,y,z]}{(x^5+y^3+z^2)}$$ where ...
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Recovering curve's equation from a given Kummer surface?

Assuming I'm not in even characteristic and that the ground field is algebraically closed for simplicity, it is known that every genus 2 curve is associated to a Kummer surface in $\mathbb P^3$. If ...
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1answer
44 views

What line bundle pulls back to the trivial line bundle

Let $X$ be an abelian surface. $C$ be a curve in $X$. Consider the projective bundle $\pi:\mathbb{P}^1_C\longrightarrow C$. This is a projective morphism. I have two questions : 1) Can we find an ...
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43 views

Algebraic independence of `Riemann-Roch' elements

First of all, I'm not too sure on what terminology should be used in the title: the question deals with the vector spaces $$\mathcal{L(D)}=\{f\colon E\to\mathbb{C} \mid f\text{ is meromorphic}, ...
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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 ...
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40 views

Finiteness of Zeros and Poles on Noetherian schemes

This exercise comes from Ravi Vakil's notes. Suppose that $X$ is an integral Noetherian scheme, and $f \in K(X)^{\times }$ is a nonzero element of its function field. Show that $f$ has a finite number ...
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globally generated but torsion cohomology?

Can a coherent sheaf on a projective variety be globally generated but have torsion higher cohomology? Is there a reference if not. Thanks...
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33 views

open subset of spec R

Suppose I have a Discrete Valuation Ring R. X = Spec R has two points, a and b with a corresponding to a residue field and b corresponding to a fraction field K. What are the non-empty open subsets 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 ...
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Question about Hartshorne's proof of Halphen's Theorem

My question comes from the proof of Theorem 6.1 in section 5.6 of Hartshorne, where I don't understand the very last step. The theorem is as follows: A curve $X$ of genus $g\geq 2$ has a nonspecial ...
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Why did Serre choose coherent sheaves?

First thing - I don't know any algebraic geometry. I'm trying to understand a little bit about quasi-coherent sheaves but not for the sake of AG, so please rely on as little knowledge as possible. ...
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Vector bundles in Ravi Vakil's notes on quasicoherent sheaves

In chapter 13 (Quasicoherent and coherent sheaves) of Ravi Vakil's wonderful notes, the author starts by discussing vector bundles, supposedly for motivation. Having understood that each locally free ...
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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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1answer
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Prove that the line $PQ$ passes through a fixed point

A right isosceles triangle $AOB$ ($O$ being the origin), is such that when $AO$ and $BO$ are extended to points $P$ and $Q$ the relation $2AP.BQ=AB^2$ holds. Prove that the line $PQ$ passes through a ...
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Quasicoherent sheaves as smallest abelian category containing locally free sheaves

On page 362 of Ravi Vakil's notes, the author says "It turns out that the main obstruction to vector bundles to be an abelian category is the failure of cokernels of maps of locally free sheaves - ...
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236 views

The case of Captain America's shield: a variation of Alhazen's Billard problem

I'm sure a lot of you are acquainted with Alhazen's Billiard problem, which involves finding the point on the edge of a circular billiard table at which a cue ball at a given point must be aimed in ...
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How to find coordinates of points on a 2D surface embedded in 3D space

kindly assist with this problem. Given an equilateral triangle in 2D plane (see figure 1) with origin (0,0) at point B, the coordinates of points A and C can be calculated as A(acos60,asin60) and ...
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45 views

Class of the variety of lines that are secant to $C$.

Let $C$ be a smooth, complex, irreducible, nondegenerate curve of degree $d$ and genus $g$ in $\mathbb{P}^3$. What is the class of the variety of lines that are secant to $C$?
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47 views

Dimension definition

Considering the definition of topological dimension of a top. Space X as max of an increasing chain of closed irreducible subsets of X. Is dimension a topological thing? I mean here, If I define ...
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47 views

Missing vital detail in the proof of codimension of singular locus (Milne's notes)

I'd like to double check something here.. If you take a look at page 175 in Milne's algebraic geometry notes (http://www.jmilne.org/math/CourseNotes/AG.pdf), there seems to be a pretty big leap from ...