The study of geometric objects defined by polynomial equations. Algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc.

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11
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2answers
293 views

Preimaging units to units

I'm interested in (unity-preserving) homomorphisms $f: S \to T$ between (commutative, with-unity) rings $S$ and $T$ so that if $f(x)$ is a unit, then $x$ was a unit to start with. For example, an ...
3
votes
2answers
644 views

Stalks of the tensor product presheaf of two sheaves

Let $(X, \mathscr{O})$ be a ringed space and $\mathscr{F}, \mathscr{G}$ be sheaves of $\mathscr{O}$-modules on $X$. Define $\mathscr{H}(U) = \mathscr{F}(U) \otimes_{\mathscr{O}(U)} \mathscr{G}(U)$. ...
5
votes
1answer
165 views

showing locally that a diagram commutes

When showing that a (natural family of) diagram of $R$-algebras for all rings $R$ commutes, why does it suffice to show that it commutes for all $R$ local with algebraically closed residue field? My ...
2
votes
3answers
148 views

Locally ringed spaces and Riemann surfaces

Let $X$ and $Y$ be Riemann surfaces, and $\mathscr{O}_X, \mathscr{O}_Y$ be the sheaves of holomorphic functions on $X$ and $Y$ respectively. It is obvious that a holomorphic map $f:X \to Y$ gives ...
5
votes
1answer
522 views

Smooth complete intersection counterexample

Does anyone know of a nice example of a non-singular complete intersection $X$ (say in $\mathbb{P}^n_k$, maybe even $k=\overline{k}$, char($k$)=0) such that it cannot be written as $E\cap H$ where $E$ ...
1
vote
1answer
213 views

Trivial line bundles

Let $X$ be a variety and $\mathbb{C}$ be the field of complex. Then $L = X \times \mathbb{C}$ is a trivial line bundle. The set of sections of this line bundle is $\Gamma(X, L)$ which consisting all ...
7
votes
0answers
158 views

The projective line over $No$ [closed]

Let $No$ be the class of surreal numbers and let us be bold and construct the projective line over $No$ as $Proj(No)$ (I don't think a "$No \times No / rel $" construction will work in case of a ...
2
votes
1answer
121 views

A divisor on a smooth curve such that $\Omega$ minus it has no nonzero sections

Let $X$ be a smooth projective curve of genus $g$. Let $\Omega$ be the sheaf of differentials. Mumford (in Abelian Varieties, sec. 2.6, in proving the theorem of the cube) asserts that there is an ...
2
votes
1answer
163 views

What are open immersions in terms of rings?

Suppose $A$ and $B$ are two commutative rings, and let $f:A\rightarrow B$ be a ring homomorphism. Suppose that the induced map on the spectra $f^*:Spec B\rightarrow Spec A$ is an open immersion. What ...
4
votes
2answers
486 views

Why is first cohomology group of divisor sheaf on riemann surface zero?

Let $X$ be riemann surface (not supposed compact) and $\mathcal D$ be sheaf of divisors on $X$.Remind that this means for $U\subset X$ open then $\mathcal D(U)$ is group of divisors on $U$. How to ...
4
votes
1answer
105 views

How to define a profinite morphism

What is the definition of a profinite morphism in http://www.math.upenn.edu/~pop/Teaching/2010_Math624/2010_Math624PS08.pdf problem 5? This is not actually a homework of mine but I was unable to find ...
4
votes
2answers
156 views

How do we describe the point 1 on the affine line?

The affine line $\mathbb{A}^1$ (over $Spec R$) is, as far as I understand, $Spec R[x]$. My guess is that $0\in \mathbb{A}^1$ is given by the (opposite of the) composite homomorphism $R[x] \to R \to F$ ...
3
votes
2answers
242 views

what are good references for learning about vector bundles and their sheaves of sections?

I am a beginner in representation theory and algebraic geometry, so that references giving clear explanations of things like the tautological line bundle on $\mathbb P^n$, its dual, and the associated ...
3
votes
1answer
283 views

Line bundles, line bundles on a homogeneous space, and sections of line bundles

I have some difficulty in understanding the concepts: line bundles, line bundles on a homogeneous space, and sections of line bundles. These concepts are on page 140 (the first paragraph of section ...
9
votes
3answers
807 views

Why doesn't Hom commute with taking stalks?

I have been learning about sheaves and am thinking about the following problem. Let $F$ and $G$ be sheaves, say of abelian groups, on a space $X$. The sheaf $Hom(F, G)$ is defined by $Hom(F, ...
0
votes
1answer
225 views

questions about coroot

I am reading the lecture notes of geometric representation theory: http://www.math.harvard.edu/~gaitsgde/267y/catO.pdf. I have a question on coroot. In general, if we have a root $\alpha$, then the ...
12
votes
3answers
1k views

Learning schemes

Could someone suggest me how to learn some basic theory of schemes? I have two books from algebraic geometry, namely "Diophantine Geometry" from Hindry and Silverman and "Algebraic geometry and ...
3
votes
0answers
238 views

question regarding Waterhouse, affine group schemes

Excerpt from Waterhouse, 14.4 Structure of Finite Connected groups. Thm. Let $A$ represent a finite connected group scheme over a perfect field of characteristic $p$. Then $A$ has the form $k[X_1, ...
2
votes
1answer
317 views

Homogeneous binary quartic forms to elliptic curves

Broadly I would like to know what is the connection between homogeneous binary quartic forms and elliptic curves. I see that the invariants on the first space under the action of $SL(2,\mathbb{C})$ ...
6
votes
1answer
238 views

How to get the connectedness theorem from the quasi-finite version of ZMT?

Let $f: X \to Y$ be a proper morphism of noetherian schemes. If the natural map $\mathcal{O}_Y \to f_*(\mathcal{O}_X)$ is an isomorphism, then a version of Zariski's main theorem states that the ...
3
votes
0answers
482 views

High resolution photos of Grothendieck [closed]

I am wondering whether there are any high resolution photos of Alexander Grothendieck available online. I did a search on Google Images, but most of the results are low quality and/or low resolution. ...
7
votes
1answer
257 views

Approximating the volume of the Jacobian of a hyperelliptic curve

For an abelian variety $A_{/\mathbb{Q}}$, its volume $vol(A(\mathbb{R}))$ appears in the conjectured Birch Swinnerton-Dyer formula for the L-series at 1. I am having trouble in understanding the size ...
3
votes
1answer
152 views

Algorithms to prove that polynomials don't have integer solutions

OK, I know that Matiyasevich's solution to Hilbert's 10th problem shows that there is no algorithm to decide whether or not a polynomial $p(x_1,\ldots,p_n)$ with integer coefficients has a solution ...
13
votes
1answer
621 views

Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be unbounded as n varies?

Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be unbounded as n varies? x,y are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of n (assuming n ...
22
votes
2answers
1k views

How many cubic curves are there?

It is well-known that there is only one "kind" of line, and that there are three "kinds" of quadratic curves (the nature of which depends on the sign of a so-called "discriminant"). It is noteworthy ...
12
votes
2answers
683 views

Algebraic versus topological line bundles

Let $X$ be a CW complex. The (isomorphism classes of) complex line bundles on $X$ are classified by the homotopy classes of maps $X \to \mathbb{CP}^\infty$, that is by the elements of $H^2(X, ...
3
votes
0answers
155 views

notes on properties of chern character of coherent sheaves

I'm trying to understand a bit about chern classes/chern characters (in the algebraic setting, for varieties over C say) and was hoping to find some notes describing some well-known properties of ch ...
7
votes
1answer
208 views

Second Chern class of a ruled surface

Let $S$ be a ruled surface over a curve of genus $g$. Is it possible to compute the second Chern class of $S$ in terms of $g$?
5
votes
2answers
608 views

Projective closure

Is the projective closure of an infinite affine variety (over an algebraically closed field, I only care about the classical case right now) always strictly larger than the affine variety? I know it ...
6
votes
4answers
264 views

Does a motive capture everything about an algebraic variety?

Is the functor from the category of projective varieties over a field $k$ to the category of pure motives over $k$, faithful? (Perhaps it is not full). Ditto: Is the functor from the category of ...
0
votes
1answer
125 views

How to find a Zariski Cover of the Grassmannian

I am wondering how to find the Zariski Cover of the Grassmannian over $\mathbb{C}$. I was hoping some reference would go through this calculation. I was recently told that it was not too hard, so I ...
17
votes
2answers
2k views

Elliptic Curves and Points at Infinity

My undergraduate number theory class decided to dip into a bit of algebraic geometry to finish up the semester. I'm having trouble understanding this bit of information that the instructor presented ...
3
votes
1answer
342 views

What is the pullback in the category of commutative algebras?

The pullback is a subset of the cartesian product in the category of commutative rings with unit. What is the pullback in the category of commutative $k$-algebras? Is it the same set as in rings?
5
votes
0answers
148 views

What is the Gauss part of Gauss-Manin connection?

The definition of Gauss-Manin connection involves de Rham cohomology. Surely, Gauss didn't work with de Rham cohomology as we know it. So, what was the context in which Gauss came up with this idea?
7
votes
3answers
647 views

Why is $\mathbb{C}[x,y]$ not isomorphic to $\mathbb{C}[x] \otimes _{\mathbb{Z}} \mathbb{C}[y]$ as rings?

I would like to know why $\mathbb{C}[x,y]$ is not isomorphic to $\mathbb{C}[x] \otimes _{\mathbb{Z}} \mathbb{C}[y]$ as rings. Thank you! 1
19
votes
1answer
470 views

What is $\operatorname{Spec}\mathbf{C}[[x,y]]/(y^{2} - x^{3} - x^{2})$?

Let $X = \operatorname{Spec} \mathbf{C}[[x,y]]/(y^{2} - x^{3} - x^{2})$. I would like to describe $X$ set-theoretically. My questions are: Can one explicitly say what the elements in $X$ are? Is it ...
4
votes
1answer
263 views

Cohomology of $\mathcal O_X$ for toric varieties

Motivated by my ignorance here, if $X$ is a projective toric variety, is $$H^m(X, \mathcal O_X) \cong \begin{cases} 0 & m > 0 \\ \mathbb C & m = 1 \end{cases} $$ as for $\mathbb ...
5
votes
2answers
135 views

Cohomological decomposition of tensor sheaves?

My question is similar to this, but not identical. I believe the following to be true, but I'd like a reference. Given (quasicoherent?) sheaves of $\mathcal O_X$ modules $E$ and $F$ on a projective ...
7
votes
1answer
124 views

System of parameters which have linear independent images in the cotangent space

Given a Noetherian, local ring $(R,m)$, can we always find a system of parameters whose images in the cotangent space $m/m^2$ are linearly independent. We can do this in the regular case, by just ...
6
votes
2answers
266 views

Status of mixed motives

From the wikipedia page: http://en.wikipedia.org/wiki/Motive_(algebraic_geometry) it appears that the category of Mixed motives $MM(k)$ over a field $k$ is still conjectural; but there is a good ...
14
votes
2answers
950 views

Theories of $p$-adic integration

What is the compelling need for introducing a theory of $p$-adic integration? Do the existing theories of $p$-adic integration use some kind of analogues of Lebesgue measures? That is, do we put a ...
3
votes
2answers
252 views

Ramanujan congruences and étale cohomology

What is a good reference for the story of congruences such as $$\displaystyle \tau(n) \equiv \sigma(11)(n) \mod\ 691$$ with a conceptual explanation with connections to étale cohomology, etc?
2
votes
1answer
224 views

the sheafification of a constant presheaf

Let X be a topological space and A be an abelian group. Give A the discrete topology. For any open set U of X, Let $\cal A(U)$ be the group of all continuous aps of U into A. Thus with the usual ...
3
votes
1answer
130 views

Different definitions of Kodaira dimension

Let X be a smooth projective variety with canonical class K. Let a be defined to be the maximum dimension of the image of X under the rational map induced by the linear system |nK| as n ranges over ...
2
votes
1answer
349 views

Proof from Reid's Undergraduate Algebraic Geometry: help with minor details

On p.28 (here) Reid proves that the curve $y^2=x(x-1)(x-\lambda)$ for $\lambda\neq 0,1$ has no rational paramaterisation. At one point (on p.29), he has the equation $$r^2=ap(p-q)(p-\lambda q)$$ where ...
0
votes
1answer
380 views

The prime spectrum in multiple variables - a basic example/question

So in algebraic geometry, the prime spectrum of the ring of polynomials over a field k is isomorphic to the set of k-homomorphisms from the ring of polynomials to k, and we do this by using the fact ...
1
vote
1answer
183 views

When does a subbase of a base generate the same topology?

Suppose that $\mathcal{B}$ is a base for a topology on a space $X$. Is there a nice way of thinking about how we can modify $\mathcal{B}$ (for instance, to simplify computations) without changing the ...
16
votes
1answer
413 views

Can an integral scheme have closed points of both positive and zero characteristic?

Background Recall that an integral scheme $X$ is a scheme which is both irreducible and reduced; equivalently, its ring of functions is an integral domain on every open subset. Given any point $p$, ...
0
votes
2answers
224 views

Are there non-noetherian topological spaces in which every open subset is quasi-compact?

I'm working through an exercise in Hartschorne concerning direct limits of sheaves on noetherian topological spaces. It seems natural to use the the fact that every open subset of a noetherian ...
3
votes
3answers
333 views

Monomorphisms of sheaves gives an injection of stalks

How do I show that a monomorphism $F \rightarrow G$ of sheaves induces an injection on stalks? When showing that monomorphism is an injection on sets one uses the maps $x \mapsto a$ and $x \mapsto ...