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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43
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3k views

Grothendieck 's question - any update?

I was reading Barry Mazur's biography and come across this part: Grothendieck was exceptionally patient with me, for when we first met I knew next to nothing about algebra. In one of his first ...
26
votes
0answers
637 views

What is the Picard group of $z^3=y(y^2-x^2)(x-1)$?

I'm actually doing much more with this affine surface than just looking for the Picard group. I have already proved many things about this surface, and have many more things to look at it, but the ...
25
votes
0answers
335 views

Geometric interpretation of the Riemann-Roch for curves

Let $X$ be a smooth projective curve of genus $g\geq2$ over an algebraically closed field $k$ and denote by $K$ a canonical divisor. I have some clues about the geometrical interpretation of the ...
25
votes
0answers
431 views

Application of Hilbert's basis theorem in representation theory

In Smalo: Degenerations of Representations of Associative Algebras, Milan J. Math., 2008 there is an application of Hilbert's basis theorem that I don't understand: Two orders are defined on the set ...
21
votes
0answers
361 views

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 ...
19
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0answers
355 views

Classification of local Artin (commutative) rings which are finite over an algebraically closed field.

A result in deformation theory states that if every morphism $Y=\operatorname{Spec}(A)\rightarrow X$ where $A$ is a local Artin ring finite over $k$ can be extended to every $Y'\supset Y$ where $Y'$ ...
19
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0answers
231 views

Homogeneous forms of degree $n$ in $n$ indeterminates over $\mathbb{Z}$: which ones come from the norm of a number field?

Is there a characterization of the homogeneous forms of degree $n$ in $n$ indeterminates over $\mathbb{Z}$ which occur as the norm of some algebraic number ring with a suitable $\mathbb{Z}$-basis? ...
18
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0answers
424 views

Is there an exposition of complex analysis firmly separating the algebra, analysis, and topology?

Complex analysis seems to work because of the interplay between algebraic geometry over $\mathbb{C}$, and analysis and topology exploiting the fact that $\mathbb{C}/\mathbb{R}$ happens to be a ...
18
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0answers
1k views

Pullback and Pushforward Isomorphism of Sheaves

Suppose we have two schemes $X, Y$ and a map $f: X\to Y$. Then we know that $\operatorname{Hom}_X(f^*\mathcal{G}, \mathcal{F})\simeq \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$, where ...
16
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0answers
189 views

What does $H^0(Y',f^*N_Y)$ measure?

Let $X$ be a smooth variety and let $Y\subset X$ be a smooth subvariety. Let $f:Y'\to Y$ be a (say, finite surjective) morphism. When $f$ is the identity, the cohomology group $H^0(Y',f^*N_Y)$ ...
14
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0answers
206 views

What is Spec of the Adeles?

Let $K$ be a global field and $A_K$ the ring of adeles. What are the prime ideals of $A_K$? I have been told that a full proof of this is quite subtle, but have been unable to find a reference ...
13
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0answers
169 views

What geometrical obstructions to $M$ being flat do elements which map to 0 in $M \otimes I$ represent?

I'm trying to get geometric intuition for the notion of a flat module over a ring, and am running into some problems with my intuition. I am comfortable with flat modules and tensor products from the ...
12
votes
0answers
384 views

Class group of $k[x,y,z,w]/(xy-zw)$

I had a homework problem (II.6.5 in Hartshorne) to compute the (Weil divisor) class group of $X=\operatorname{Spec} k[x,y,z,w]/(xy-zw)$. I have accomplished this; however, I used some results I don't ...
11
votes
0answers
224 views

Prop. 3.2.15 in Liu: Geometrically reduced algebraic variety

I have a problem with proposition 3.2.15 of Algebraic geometry and arithmetic curves of Qing Liu: Let $X$ be an integral algebraic variety over $k$. If $X$ is geometrically reduced then $K(X)$ is a ...
10
votes
0answers
86 views

Irreducibility of cyclotomic polynomials via schemes

A few months ago, someone told me there existed a scheme theoretic proof of the irreducibility of cyclotomic polynomials. I've tried coming up with a proof, and when that didn't really yield anything ...
10
votes
0answers
309 views

Visualizing a Calabi Yau

I would like to understand how I can visualize the quintic threefold $$ z_1^5 + z_2^5 + z_3^5 + z_4^5 +z_5^5 - 5\psi z_1z_2z_3z_4z_5 = 0$$ For a similar problem, Hanson proposes the following: ...
10
votes
0answers
319 views

Self-Intersection Number $-2$

I am new here, but hopefully you can help me with a concrete problem I have. I try to compute a Self-Intersection Number of a constructed curve in an analytic surface. I know the answer by some ...
10
votes
0answers
410 views

Proof of residue theorem (residue formula) for differential forms on curves over an arbitrary closed field.

I have been reading the book Algebraic Geometric Codes: Basic Notions by Tsfasman, Vladut and Nogin. They give a residue formula like this: Let $\mathbb{k}$ be an algebraically closed field and $X$ ...
10
votes
0answers
193 views

Curves and Sums-of-Powers Representations

Jacobi first noticed the connection between the functions that bear his name and counting the representations of sums-of-squares, \begin{eqnarray} \theta_{3}^{n}(q) = \left( \sum_{k \in \mathbb{Z}} ...
9
votes
0answers
167 views

Understanding Bertini's theorem

Let's suppose that I am given a pencil generated by the vector fields $X$ and $Y$ in $\mathbb{C}^2$, $\{ Z_\lambda \}_{\lambda\in\mathbb{P}^1}$, that is, $$ Z_\lambda = X + \lambda Y $$ Assume that ...
9
votes
0answers
184 views

Reality check: $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(1)$ and $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(-1)$ are both Möbius strips?

This question follows up this previous question, which has an accepted answer that I am having trouble believing. (Edit: the answer has now been fixed; thanks Georges!) I have been working on giving ...
9
votes
0answers
646 views

Arithmetic and geometric genus

There are two notion of genus in algebraic geometry, namely arithmetic genus $p_a=(-1)^{\dim X}(\chi(\mathcal{O}_X)-1)$ and geometric genus $p_g=\dim H^0(X,\Omega^{\dim X})$. I keep forgetting ...
9
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0answers
149 views

(Weil divisors : Cartier divisors) = (p-Cycles : ? )

Suppose $X$ verifies the suitable conditions in which Weil (resp. Cartier) divisors make sense. The group of Weil divisors $\mathrm{Div}(X)$ on a scheme $X$ is the free abelian group generated by ...
9
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0answers
335 views

Trivial intersection of algebraic sets?

The question came up while reading a bit more into the Hilbert-Zariski theorem I asked about the other week. Suppose $V$ is an algebraic variety over arbitrary field $k$. (For this situation, I'll ...
8
votes
0answers
74 views

Why is the Leibniz rule a sufficient ingredient in the construction of the tangent space?

This is a very soft question, but I am wondering if anyone can shed light on why it is that the product rule (and linearity) provide exactly the right requirement for the space of derivations to be ...
8
votes
0answers
160 views

Global sections of vector bundles on a complex elliptic curve and analytic functions

Let me fix an elliptic curve $E$ over complex numbers with distinguished point $x \in E$. Thanks to Atiyah we know everything about discreet parameters of vector bundles and its moduli spaces. But I ...
8
votes
0answers
157 views

algebraic $1$-forms vs analytic $1$-forms

First let's fix some definitions: Definitions: Complex manifold (of dimension n): Is a locally ringed space $(X,\mathscr F)$, where there is an open cover $\bigcup_{i\in I} U_i=X$ such that ...
8
votes
0answers
102 views

Is there anything to be learned from the spectrum of a cohomology ring?

Given some topological space, $X$, is there any benefit to studying $Spec(H^*(X))$, or is everything we care about already available "in the algebra"? As $H^*$ is a graded ring, does this question ...
8
votes
0answers
140 views

Canonical sheaf not globally generated for a certain surface.

Me and a friend tried the following problem, but with no luck. Anything would be appreciated: Let $X \rightarrow S$ be an arithmetic surface such that for some $s \in S$, $X_s$ is the union of two ...
8
votes
0answers
221 views

Why do these two constructions of a sheaf associated to a module on a projective scheme agree?

Let $B$ be a graded ring an $M$ be a $\mathbb Z$-graded $B$-module. We can associate to $M$ a sheaf of modules on $\operatorname{Proj} B$ by defining the sheaf on the principal open sets $D_+(f)$ to ...
8
votes
0answers
162 views

Transformations that map points inside the sphere to points inside the sphere

I am trying to figure out what is the most general linear transformation that maps points inside the unit sphere to points inside the unit sphere. I am slightly abusing the word linear here by ...
8
votes
0answers
204 views

Ideal of the pullback of a closed subscheme

Let $f : X \to Y$ be a morphism of schemes and $J \subseteq \mathcal{O}_Y$ a quasi-coherent ideal. Let $I$ denote the image of $f^* J \to f^* \mathcal{O}_Y = \mathcal{O}_X$. Then $I \subseteq ...
8
votes
0answers
129 views

Tropical-like redefinitions of addition and multiplication?

I've been impressed at the rich structure of tropical mathematics, all consequences of the seemingly mundane starting point of replacing classical $a+b$ by $\min(a,b)$ (or max), and replacing ...
8
votes
0answers
163 views

Tensoring is thought as both restricting and extending?

I hope these questions are not too trivial. Let $I$ be an ideal in $R$. Write $I'\subseteq R[t]$. Then the notion of tensoring $$ (R[t]/I')\otimes_{\,\mathbb{C}[t]} \mathbb{C}[t]/\langle t-c ...
8
votes
0answers
490 views

Prerequisite of Projective Geometry for Algebraic Geometry

I studied Euclidean Geometry in high-school, and I have not studied anything relates to geometry since I started studying in university. I am now intending to study Algebraic Geometry, however, I ...
7
votes
0answers
173 views

Smooth subvariety at smooth points

Let $f: X \to Y$ be a finite, surjective morphism of smooth, quasi-projective varieties over a field $k$ of characteristic zero. Let $p \in X$. Is it true that I can find a subvariety $X' \subseteq X$ ...
7
votes
0answers
93 views

What does the space of non-diagonalizable matrices look like?

Let $k$ be a field (I would be happy working entirely over $\mathbb C$). Consider the action of $G=GL_n(k)$ by conjugation on the set of $n\times n$ matrices over $k$. The collection $X$ of matrices ...
7
votes
0answers
313 views

Trying to Understand Lefschetz Pencils

I'm reading on Lefschetz pencils, and I'm trying to understand condition ii) better, tho I would appreciate insights on condition i), and in general. A Lefschetz pencil on a $4$-manifold $X$ is a ...
7
votes
0answers
126 views

Why do algebraic varieties contain curves passing through two given points

Let $X$ be an algebraic variety over the complex numbers. My definition of an algebraic variety is a finite type separated $\mathbf C$-scheme. Someone told me that such varieties have the following ...
7
votes
0answers
113 views

On an exercise from Hartshone's Algebraic Geometry, Ch I sect 4

My question is about the Ex. 4.9 page 31 in the book GTM52 by Robin Hartshone. Let $X$ be a projective variety of dimension $r$ in $\mathbb{P}^n$, with $n\geq r+2$. Show that for suitable choice ...
7
votes
0answers
124 views

Surfaces ruled over elliptic curves

Ground field $\Bbb{C}$. Algebraic category. Elliptic surfaces are those surfaces endowed with a morphism onto some smooth curve, with generic fiber an elliptic curve. Suppose $E$ is an elliptic ...
7
votes
0answers
142 views

Invertible rational functions

I am looking for references for the following facts. ...
7
votes
0answers
156 views

Divisor class group of an affine surface

In this topic the OP considers the following surface: $X=\mathcal{Z}(z^3-y(y^2-x^2)(x-1))$. (The field it's not explicitely mentioned, but for geometric reasons this can be algebraically closed.) He ...
7
votes
0answers
147 views

Poincaré duality and intersection

Let's take $X$ and $Y$ K3 surfaces and $Z\subset X\times Y$ an algebraic cycle of dimension 2. I know that the Poincarè dual of $Z$, namely $[Z]$, is in $H^4(X\times Y,\mathbb{Z})$ and by Kunneth ...
7
votes
0answers
125 views

Two cones over a projective variety

Let $A$ be a commutative graded algebra over a field $k$ and $X=\operatorname{Proj}(A)$ is a smooth scheme, then $E=\oplus_{i \geq 0} \mathcal O_X(i)$ is a quasi-coherent sheaf of algebras on $X$. I ...
7
votes
0answers
111 views

Continuous choice of basis for subspaces

Consider the flag variety (or flag manifold, depending on who you are) $V=\mathrm {Fl} (3,\mathbb C)$ of complete flags of subspaces of $\mathbb C^3$. That is, an element of M is a tuple (L , P) ...
7
votes
0answers
221 views

Do Neron models of hyperbolic curves exist

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$. Does there exist a Neron model $\mathcal X$ for $X$ over $O_K$? By a Neron model, I mean ...
7
votes
0answers
469 views

A problem about the twisted cubic

I have some difficulty with the following problem: Let $f : k → k^3$ be the map which associates $(t, t^2, t^3)$ to $t$ and let $C$ be the image of $f$ (the twisted cubic). Show that $C$ is an ...
7
votes
0answers
236 views

$\mathcal{L}$ is very ample, $\mathcal{U}$ is generated by global sections $\Rightarrow$ $\mathcal{L} \otimes \mathcal{U}$ is very ample

Let $\mathcal{L},\mathcal{U}$ be invertible sheaves over a noetherian scheme $X$, where $X$ is of finite type over a noetherian ring $A$. If $\mathcal{L}$ is very ample, and $\mathcal{U}$ is generated ...
7
votes
0answers
162 views

What's the relation between cohomology and unramified Galois covering of curves

The following statement in a paper puzzles me: "We may view $H^1(X(N), \mathbb{Z}/\ell\mathbb{Z})$ as classifying unramified Galois coverings of $X(N)$ with structure group ...