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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2
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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 ...
2
votes
1answer
38 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 ...
7
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0answers
204 views
+100

Weighted projective space and $\mathrm{Proj}$

I'm trying to solve a problem from Jenia Tevelev's notes on GIT. (Problem 5 at the end of this pdf.) Compute $$\operatorname{Proj}\frac{\mathbb{C}[x,y,z]}{(x^5+y^3+z^2)}$$ where ...
1
vote
0answers
20 views

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 ...
2
votes
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 ...
2
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0answers
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}, ...
2
votes
0answers
56 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 ...
1
vote
0answers
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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0answers
25 views

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...
0
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0answers
34 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 ...
0
votes
0answers
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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0answers
40 views

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 ...
4
votes
2answers
159 views

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. ...
3
votes
0answers
40 views

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 ...
5
votes
1answer
70 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 ...
2
votes
1answer
44 views

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 ...
7
votes
0answers
56 views

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 - ...
8
votes
1answer
242 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 ...
-2
votes
0answers
28 views

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 ...
0
votes
1answer
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 $C$?
1
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2answers
48 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 ...
0
votes
0answers
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 ...
4
votes
0answers
93 views

Is it normal surface of general type to have infinitely many positive rank elliptic curves?

I am not good at algebraic geometry and almost surely am misunderstanding something. Got an alleged argument against Bombieri-Lang conjecture and would like to know what the mistake is. One of the ...
1
vote
0answers
11 views

Prove that $\{m \in S_{\sigma} \, | \, \gamma(m) \neq 0\}$ is a face of $\sigma^V \cap M$

I am trying to solve exercise 3.2.6 pag.124 of Cox, Little, Schenck book http://www.math.colostate.edu/~renzo/teaching/Toric14/CoxLittleShenck.pdf because it is required to prove orbit-cone ...
0
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0answers
45 views

Proof Algebraic geometry R. Hartshorne

In book Algebraic geometry R. Hartshorne I'm confused in the proof of theorem 3.4 first chapter. 1) In the first paragraph of the proof the theorem 3.4: what is the significance of "this isomorphis ...
0
votes
0answers
74 views

What is an algebra and what is it's representation?

Heyho, i've kind of got an understanding problem what exactely an algebra and especially it's representation is. In my case it is said, that the relation $R_{12}(u-v) (L(u) \otimes \mathrm{I}) \; ...
2
votes
0answers
64 views

What is the push forward of the canonical class?

Let $X$ be an abelian surface over $\mathbb{C}$. And let $i:X\longrightarrow X$ be the inverse map. $i$ is a degree 2 morphism. We consider $Y$ the quotient of $X$ by the action of $i$, that is, ...
1
vote
1answer
43 views

Proving $\gcd(f_i)=1\Rightarrow \mathbb{A}_\mathbb{C}^n\setminus \{f_i\}$ is not affine

I need to prove the following lemma: Lemma: Let $f_i\in \mathbb{C}[x_1,\dots,x_m]$ s.t. $\gcd(f_1,f_2,\dots,f_n)=1\quad(1<n\le m)$. Prove that the variety ...
0
votes
1answer
39 views

Question about the degree of a morphism

Suppose that $\phi$ is a morphism between compleax algebraic varieties named $X$ and $Y$. I know that the degree of the morphism $\phi= [Rat(X):Rat(Y)]$. Suppose that $\phi$ is a one degree morphism. ...
1
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0answers
27 views

subsheaf of free sheaves

Let $X$ be an irreducible nodal curve, $E:=\oplus_{i=1}^r \mathcal{O}_X$ be a free sheaf on $X$ and $F \subset E$ a (coherent) subsheaf. Is it possible to write $F$ as a direct sum of subsheaves ...
5
votes
0answers
49 views

What is the obstruction to extending a linear map on tangent spaces of a variety to a regular map on neighborhood?

Suppose that $X$ and $Y$ are algebraic varieties of the same dimension $n$. If $p$ and $q$ are points in $X$ and $Y$ respectively, suppose that there is a linear map $i : T_p X \to T_q Y$. My vague ...
6
votes
1answer
88 views

First sheaf cohomology $H^1(\mathscr{O}_D, \mathbb{D})=0$

Can I get a hint on this problem? Given a finite divisor $D=p_1+\dots +p_m -q_1 -\dots -q_n$ on the unit disk $\mathbb{D}$, how do I show that the first sheaf cohomology group $H^1(\mathscr{O}_D, ...
3
votes
1answer
62 views

What's the kernel of the codiagonal $k[x] \otimes_k k[x] \rightarrow k[x]$?

maybe this question is really stupid, but I could not solve it after thinking for a while. Let $I$ be the kernel of the codiagonal map $$k[x] \otimes_k k[x] \rightarrow k[x]$$ given by $f(x) \otimes ...
1
vote
0answers
27 views

Riemann-Roch for nodal curves

Let $X$ be an irreducible, nodal curve and $E$ a coherent subsheaf of a free sheaf $\oplus_{i=1}^r \mathcal{O}_X$ on $X$ of rank strictly less than $r$. Assume that $r \ge 2$. It follows that $H^0(E)$ ...
4
votes
0answers
46 views

Morphism between surfaces

Suppose that $S$ is a surface of general type. Let $K_S$ the canonical bundle of $S$ and $\phi=\phi_{K_S}$ the canonical map. Suppose that the canonical map is a morphism from $S$ to ...
1
vote
0answers
14 views

Calculate rotation matrix starting with 8 coordinates from a box to an axis-aligned box

I've got a rectangular box that's described by the coordinates of its 8 corners. Now I want to calculate the rotation matrix which would rotate this box so its edges align with the coordinate system. ...
1
vote
0answers
30 views

How can I blow-up a smooth projective surface with certain conditions?

Let $X$ be a smooth projective surface and $K$ a canonical divisor on $X$. Suppose $V$ and $W$ are subspaces of $H^0(X, \mathcal{O}_X(nK))$ (for $n$ large). Q: How can we blow-up $X$ to obtain $\pi: ...
1
vote
1answer
49 views

Degree of a projective variety

Let $X \subset \mathbb{P}^n$ be a projective variety of dimension $k <n$. By an equivalent definition of dimension, $k$ is the smallest integer such that there exists an open set of $G(n-k-1,n)$, ...
0
votes
0answers
41 views

Quasicoherent-sheaves and pushfoward

How to prove the proposition in the picture below? It seems to be easy, but I am a beginner. Thanks in advanced for your help!
2
votes
0answers
27 views

Question on the matrix of a Kaehler Metric in Normal Coordinates

I am currently studying normal coordinates on a Kaehler manifolds: Let $h$ be a Kaehler metric on a complex manifold $M$ and let $p \in M$. Let $(z_1,..,z_n)$ be a coordinate chart such that $h$ is a ...
4
votes
1answer
55 views

degree of an etale cover of the affine line

Let $X\subset \mathbb{A}^N_k$ be an irreducible smooth variety over an algebraically closed field $k$. Suppose we have an etale map $\pi:X\to \mathbb{A}^1_k$. Are there any bounds on the degree of ...
2
votes
0answers
75 views

Can someone give me the spherical equation for a 26 point star?

This is the object that I am trying to find the volume of. This can be treated as a "26 point star". What I need is an equation to describe it. If anyone has that surface in spherical ...
2
votes
1answer
30 views

Two questions about Schubert calculus and Schur functions.

I am reading the file. I have a question on pae 28. How to prove that $[X_{\{2,4\}}] = S_{(1)} = x_1 + x_2 + \cdots$ and $S_{(1)}^4 = 2 S_{(2,2)} + S_{(3,1)} + S_{(2,1,1)}$? I tried to verify ...
2
votes
0answers
34 views

Schubert calculus and number of lines satisfying some properties.

I am reading the file. I have a question on pae 18. It is said that: Given a line in $\mathbb{R}^3$, the family of lines intersecting it can be interpreted in $G(2, 4)$ as the Schubert variety $$ ...
2
votes
1answer
53 views

Can one have a nontrivial 'resolution of singularities' of a smooth variety?

Suppose $z_1,z_2$ are coordinates on $\mathbb{A}^2$ and $(w_1,w_2)$ homogeneous coordinates on $\mathbb{P}^1$. We can define a subvariety $X \subset \mathbb{A}^2 \times \mathbb{P}^1$ by $w_1z_2 - ...
2
votes
0answers
37 views

Detail regarding tangent spaces and dual varieties from Harris's Algebraic Geometry: A First Course

In Harris's Algebraic Geometry: A First Course, Example 16.20, the author shows that the dual of the dual variety $X^{*}$ is the original variety $X$. I think in chapter 15, Harris mentions that he'll ...
1
vote
1answer
44 views

Question about Normal Coordinates on a Kaehler Manifold

I am currently reading FY Zheng's textbook, "Complex Differential Geometry". In section 7.4 Proposition 7.14, he is trying to prove thata metric $h$ Kaehler is equivalent to the statement, "For any $p ...
3
votes
1answer
59 views

Non-linear equivariant maps between group representations

Given two representations $\pi_1$ and $\pi_2$ of a group $G$ (let's say it's a compact Lie group), a natural thing to study are linear equivariant maps A between them: $$ A \pi_1 = \pi_2 A $$ I'm ...
9
votes
0answers
95 views

Can you integrate on a scheme?

As the question suggests, can you integrate on a scheme? How? I don't even know if this is even a well-posed question...
1
vote
0answers
30 views

Re-write the Riemann Roch theorem

How can I write the Riemann-Roch Theorem with the following definitions: Definition $1$: $\mathcal{F}$ is an scheme of fractional ideals on $C$ if is coherent and for all $P\in C$, the stalk ...