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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61
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+500

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 ...
41
votes
0answers
885 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 ...
32
votes
0answers
895 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 ...
30
votes
0answers
574 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 ...
25
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0answers
907 views

On the problem of polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$

The question titled "Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$" which was posed on MathOverflow attracted quite a lot of attention (and may be the question with most wrong ...
25
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0answers
460 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 $Y'$ ...
24
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0answers
2k 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 ...
21
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0answers
345 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 ...
20
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220 views

Why are period integrals naïve periods?

Apologies for the long question. I recall the definition of a (naïve) period according to Kontsevitch and Zagier [KS]: A (naïve) period is a complex number whose real and imaginary parts are ...
20
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220 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 ...
19
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213 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 $H^0(Y',f^*N_Y)$ ...
18
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259 views

Is there a proof of Bézout's theorem via residue theory?

Let's define intersection numbers as follows. Consider a collection $f_1,\dots, f_n$ of holomorphic functions on some neighborhood of zero in $\mathbb C^N$ cutting out divisors $D_1$, all of which ...
17
votes
0answers
342 views

Intersection of twisted cubics in $\mathbb{P}^3$

Suppose we have two twisted cubics $C_1$, $C_2$ in $\mathbb{P}^3$ such that both of them lie in some cubic surface, which means that $h^0(\mathbb{P}^3, I_{C_1\cup C_2}(3))>0$. I want to show that ...
16
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0answers
381 views

Wanted: A purely algebraic proof of the Frobenius theorem on distributions

Is there a purely algebraic proof of the Frobenius theorem? Here's a rough sketch of what i'm looking for: Let $Der(R)$ denote the $R$-module of ($R$-valued) derivations of the algebra $R$ endowed ...
16
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0answers
341 views

2 smooth algebraic varieties, isomorphism $\mathcal{D}_{X \times Y} \to p_1^* \mathcal{D}_X \otimes p_2^*\mathcal{D}_Y$ as sheaves of rings?

Let $X$ and $Y$ be two smooth algebraic varieties. How do I see that I have an isomorphism$$\mathcal{D}_{X \times Y} \overset{\sim}{\to} p_1^* \mathcal{D}_X \otimes p_2^*\mathcal{D}_Y$$as sheaves of ...
13
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0answers
701 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 ...
12
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131 views

Intuitive/geometric way of thinking about effective divisors?

What is the motivation/intuition/geometric way of thinking about an effective divisor? I know that a divisor is effective if all its coefficients are non-negative. We write $D \ge 0$ for ...
12
votes
0answers
268 views

Maximum number of intersection points of two different Bernoulli lemniscates

What is the maximum number of intersection points of two different Bernoulli lemniscates in the real plane? (Of course two identical lemniscates share an infinite number of points.) Here are ...
12
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0answers
664 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 $X$ ...
11
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233 views

Has anyone seen this combinatorial identity involving the Bernoulli and Stirling numbers?

Does anyone know a nice (combinatorial?) proof and/or reference for the following identity? $$\left( \frac{\alpha}{1 - e^{-\alpha}} \right)^{n+1} \equiv \sum_{j=0}^n \frac{(n-j)!}{n!} |s(n+1, ...
11
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0answers
236 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 ...
11
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0answers
275 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 ...
11
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515 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: ...
11
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0answers
456 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 ...
11
votes
0answers
1k 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 ...
11
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0answers
676 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 ...
10
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89 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 - ...
10
votes
0answers
117 views

Study of rings of the form $R+I$

In my life I saw lots of ways of constructing rings: polynomial rings, quotient rings, localizations, endomorphism rings, rings of fractions, integral closure of a ring, center of a ring, etc... These ...
10
votes
0answers
127 views

Multiplicity of point as a zero of polynomial.

Let $C \subset \mathbb{P}_2$ be a projective curve defined by a homogeneous polynomial $P(x, y, z)$, and let $L \subset \mathbb{P}_2$ be the line $\{z = 0\}$. Assume $L$ is not a component of $C$. If ...
10
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0answers
108 views

Non-Universal Delta Functors

Recall a delta functor on an abelian category is a collection of functors $H^n$, $n \ge 0$ and connecting homomorphisms associated to each short exact sequence so that we get a corresponding long ...
10
votes
0answers
263 views

Hartshorne Theorem 8.17

I can't understand the proof of theorem 8.17 from Hartshorne's "Algebraic Geometry". Namely, he says that we have an exact sequence $$ 0 \to \mathcal J'/\mathcal J'^2 \to \Omega_{X/k} \otimes ...
10
votes
0answers
168 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 ...
10
votes
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215 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}} ...
10
votes
0answers
331 views

In which commutative algebras does any derivation possess a flow?

Suppose $A$ is a commutative algebra over $\mathbb{R}$ with unity. $\mathbb{R}$-linear map $\xi\colon A\to A$ is a derivation of $A$ iff $\xi(ab)=a\xi(b)+\xi(a)b$ for any $a,b\in A$. If $\gamma\colon ...
9
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0answers
136 views

Explicit examples of (co)limit arguments in other fields

Over the past weeks, I have noticed that high level lecture notes in subjects like algebraic geometry, algebra, and algebraic topology often sketch proofs in the following form: Proof sketch ...
9
votes
0answers
108 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...
9
votes
0answers
149 views

Algebraic methods to compute the cohomology ring of the complex topology of a variety?

Suppose $V$ is an affine (resp. projective) subvariety of the affine (resp. projective) space $\mathbb A_\mathbb C^n$ (resp. $\mathbb P_\mathbb C^n$) with vanishing ideal $I\subseteq\mathbb ...
9
votes
0answers
167 views

Calculation with Leray spectral sequence

The Leray spectral sequence is a cohomological spectral sequence of the form $$H^p(Y;R^q f_*(F)) \Longrightarrow H^{p+q}(X;F)$$ for abelian sheaves $F$ on a site $X$ and morphisms of sites $f : X \to ...
9
votes
0answers
99 views

Relationship between twisting sheaves and divisor sheaves

I'm not really entirely sure how to think about Serre's twisting sheaves $\mathscr{O}(i)$ - on any $\text{Proj}$ construction, really, but let's stick to something like $\mathbb{P}_2$ for now for ...
9
votes
0answers
179 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 ...
9
votes
0answers
194 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
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0answers
262 views

When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ for $\Gamma$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least ...
9
votes
0answers
178 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 ...
9
votes
0answers
409 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 ...
9
votes
0answers
294 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 ...
8
votes
0answers
125 views

When can stalks be glued to recover a sheaf?

Let $\mathcal{F}$ be a sheaf over some topological space. The stalks are $\mathcal{F}_x= \underset{{x\in U}}{ \underrightarrow{\lim}} \mathcal{F}(U)$. Is there a special name for a sheaf that ...
8
votes
0answers
64 views

Can every variety appear as singular locus?

let $V$ be a projective variety. Does there always exist another variety $X$ such that $\operatorname{Sing}X=V$? Here $\operatorname{Sing}X$ means the singular locus of $X$ with reduced structure.
8
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90 views

Finding irreducible components of Spec$(R/I^n)$

Let $R= k[x,y,z]/(xy,yz,zx)$. Let $I=(x)$. What are the irreducible components of $\mathrm{Spec}(R/I^n)$ where $n \geq 2$ and $k$ is a field? For solving this problem I'm trying to use following ...
8
votes
0answers
88 views

Genus of $k(T)$?

Let $F$ be a function field in one variable with total constant field $k$, let $X$ be the set of all places $F$, and let $S$ be a nonempty finite subset of $X$. Then the genus of $F$ is equal to the ...
8
votes
0answers
195 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 $X' \subseteq X$ ...