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. ...

learn more… | top users | synonyms (1)

3
votes
1answer
485 views

Presheaf which is not a sheaf — holomorphic functions which admit a holomorphic square root

I'm thinking about a problem in Ravi Vakil's algebraic geometry notes http://math.stanford.edu/~vakil/216blog/ (Exercise 3.2B) and I'm having trouble with the second part of the exercise as my ...
3
votes
1answer
319 views

Divisors of rational functions on curves at singular points

Suppose $C$ is an algebraic curve (which has singular points) over an algebraically closed field $k$, and that $f$ is a rational function on $C$. How does one defines the Weil divisor of $f$? The ...
3
votes
2answers
244 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
344 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?
3
votes
2answers
193 views

How to find a bijection between R-valued points of X and local ring homomorphisms?

I am trying to prove the following fact for a homework assignment in algebraic geometry: Let R be a local ring, and X a prescheme. Show that there is a one-one correspondence between R-valued points ...
3
votes
2answers
281 views

To what extent do the stories on manifolds carry over to schemes?

This is a follow-up (refinement?) of this question. In learning some algebraic topology, I've learned to think of an affine scheme as spec $R$. (I've been told that this is a legitimate use of ...
3
votes
2answers
231 views

Flatness of local rings

What do I miss in the following? Let $R$ be a commutative Noetherian ring with unit. A map $f:M\to N$ of $R$-modules is injective/surjective iff the associated map $f_p:M_p\to N_p$ on the ...
3
votes
1answer
50 views

Hartshorne Ex III5.7 c)

Suppose $X$ is a reduced proper schemes over a noetherian ring, $X_i$ are its irreducible components,$L$ is a invertible sheaf on $X$. If $L|_{X_i}$ are all ample, how do we show $L$ is ample? If ...
3
votes
1answer
95 views

sheafification definition?

I came across two different definitions of sheafification and I'm not sure how they are equivalent. One of them is here: About the sheafification Another one is from Tennison's sheaf theory: Given a ...
3
votes
2answers
64 views

Bijection between solutions of polynomials equations and spectrum of the quotient ring.

I would like to see the connection between the set of solutions of a system of polynomial equations and a spectrum of the quotient polynomial ring. Given $f_1, f_2,\cdots, f_m \in k[x_1,x_2,\cdots, ...
3
votes
1answer
49 views

Two questions on the definition of $\mathcal{O}_X(U)$ for an affine scheme $X$.

Let $X=\operatorname{Spec}(A)$ be an affine scheme. Hartshorne defines $$ \mathcal{O}_X(U)=\{s\colon U\to\coprod_{\mathfrak{p}\in U} A_\mathfrak{p} \mid s(\mathfrak{p})\in A_\mathfrak{p} \text{ and } ...
3
votes
1answer
38 views

Adjunction counit for sheaves is isomorphism

Let $f\colon X \to S$ be a proper morphism of varieties over $\mathbb{C}$ with $f_* \mathcal{O}_X = \mathcal{O}_S$ and $\mathcal{G}$ be a coherent sheaf on $S$. Then we have a natural morphism ...
3
votes
2answers
76 views

Why do only fixed points contribute to the Euler characteristic?

Let $G$ be an algebraic group with zero Euler characteristic, acting on a variety $X$ (over $\mathbb C$). I read some time ago that then the Euler characteristic of $X$ can be computed as ...
3
votes
1answer
77 views

Torsion sheaves on a curve

This is probably a silly question, but I'm a bit confused. Regarding exercises 6.11 and 6.12 of Chapter II of Hartshorne: Let $X$ be a nonsingular projective curve over an algebraically closed field ...
3
votes
1answer
80 views

Why is it called “elliptic” curve?

One of my favourite and most studied algebraic curve is the elliptic curve. But something that I have never asked myself is: Why do they call this nonsingular cubic curve an "elliptic" curve? ...
3
votes
1answer
66 views

rank of quadrics

Consider the quadric $xw-yz$ in $\mathbf{P}^3$ (all over $\mathbf{C}$), and the Klein quadric $x_0 x_5+x_1 x_4+x_2 x_3$ in $\mathbf{P}^5$. I want to determine the rank of these quadrics. For the first ...
3
votes
1answer
41 views

Degree of ample bundle over projective curve is positive

(From Vakil's notes, Exercise 18.4.K) If $C$ is an integral projective curve over a field $k$, and $\mathscr{L}$ is an ample line bundle on $C$, why is the degree of $\mathscr{L}>0$? If $C$ is ...
3
votes
1answer
55 views

local parameter on an irreducible affine algebraic curve

On page 14 of Shafarevich's Basic Algebraic Geometry 1, it is stated that for an irreducible affine algebraic curve $X: f(x,y) = 0$, and a nonsingular point $P \in X$, there is a regular function $t$ ...
3
votes
2answers
45 views

Reid's UAG problem 4.7: isomorphism of affine line with a curve

Let $C:$ $(Y^2=X^3+X^2)\subset \mathbb{A}^2$; the familiar parametrization $$ \varphi\colon \mathbb{A}^1 \to C,$$ given by $$ T \mapsto (T^2-1,T^3 -T)$$ is a polynomial map, but is not an ...
3
votes
1answer
29 views

Geometric meaning of the splitting field over a function field

Let $K$ be a field and consider the ring of polynomial in two variables $K[x,t]$. Now take a polynomial $f(x)\in K[x]$ of positive degree and consider it in the bigger ring $K(t)[x]$. Suppose that ...
3
votes
3answers
79 views

Show that a smooth plane quartic is never hyperelliptic

I have been asked to show that a smooth plane quartic is never hyperelliptic. I know that i) The genus of any such curve is 3 ii) The statements of Riemann-Roch and Riemann-Hurwitz iii) A curve is ...
3
votes
1answer
75 views

A doubt in the proof of Prop. 1.10 of Hartshorne's Algebraic Geometry

I have a doubt in the proof of Proposition 1.10 of Hartshorne's book Algebraic Geometry, which states that if $Y$ is a quasi-affine variety, then its dimension is the dimension of its closure. In ...
3
votes
2answers
173 views

Sites or youtube videos to learn algebraic geometry

Is there any sites or free lecture videos to learn algebraic geometry? or should I call abstract algebra? I want to understand about rings, ideals, and real spectrum of rings but my understanding on ...
3
votes
1answer
47 views

functoriality of $K(G,1)$ spaces in a particular situation involving complex elliptic curves

I apologize if the subject doesn't accurately describe my question. Let $F_2$ denote the free group on two generators. Suppose you have some group homomorphism $A : ...
3
votes
1answer
59 views

Hom functor of quasi-coherent sheaf maybe not quasi-coherent

I notice that some books say that for arbitrary quasi-coherent sheaves $F$, $G$ over a scheme $X$, the $\mathcal O_X$-module $\mathrm{Hom}_{\mathcal O_X}(F,G)$ maybe not quasi-coherent, who can give ...
3
votes
1answer
39 views

Showing the set of diagonalizable matrices is constructible

Identifying $M_n(k)$ with $k^{n^2}$ with $k$ algebraically closed, I am asked to show that the subset of diagonalizable matrices, $D_n$ is constructible. Constructible is defined as being the finite ...
3
votes
1answer
37 views

A scheme with not-numerable affine covering.

This question maybe finds the answer in the definition. Let $X$ be a scheme with not hypothesis on it (not noetherian, not affine,... ). Can I find a scheme that has a not numerable affine covering? ...
3
votes
1answer
42 views

Structure sheaves in different point are not isomorphic?

Suppose $X$ is a smooth projective variety over $\mathbb{C}$. How can one understand that in $D(Coh(X))$, the structure sheaves corresponding to different points of $X$ are all non-isomorphic? Here by ...
3
votes
1answer
44 views

Isomorphism of Grassmannians

I want to prove that two CW complexes $\mathrm{Gr}_{n}(\mathbb{R}^{n+k})$ and $\mathrm{Gr}_{k}(\mathbb{R}^{n+k})$ are isomorphic to one another. I'm pretty sure I can just show that the number of ...
3
votes
2answers
69 views

The Affine Tangent Cone

I'm failing to see how exactly is the tangent cone at a singular point on a curve picking out all the different tangent lines through this singular point (say the origin in $\mathbb{A}^2$)? Could ...
3
votes
1answer
98 views

What is the normalization of the ring $\mathbb C[x,y,t]/(t^3-x^3y)$?

I would like to compute the normalization of the ring $A=\mathbb C[x,y,t]/(t^3-x^3y)$, but I do not know how to proceed. I am not an expert in normalizations, and the only examples I saw were ...
3
votes
1answer
92 views

How to prove the sheafification is a sheaf?

I know that this question might be too easy for you, but I have to study on my own, so please explain for me. In the page 64, Hartshone defined the sheafification of a presheaf $\mathcal{F}$ by ...
3
votes
1answer
53 views

Ensuring I have a closed point

Hartshorne, Algebraic Geometry, Exercise II.3.20, reads (in part): Let $X$ be an integral scheme of finite type over a field $k$. (a) [Prove:] For any closed point $P \in X$, $\dim X = \dim ...
3
votes
1answer
29 views

Proof of Castalnuovo's rationality criterion

Let $S$ be a complex projective smooth surface. If $D$ is a divisor on $S$, let's write $h^i(D)$ for $dim H^i(S,\mathcal{O}_S(D))$, where $\mathcal{O}_S(D)$ is the invertible sheaf associated to $D$. ...
3
votes
1answer
42 views

Computing the closed subschemes of the projective line over a field

(Specifically, this is III-15 in E&H, but I feel like I've hit a brick wall in actually applying the definitions they've given to this example.) In Chapter I of The Geometry of Schemes, E&H ...
3
votes
1answer
128 views

Number of common zeros of two quadratic polynomials in ${\Bbb C}[t,x]$

The following theorem is in Artin's Algebra(2nd edition): Theorem 11.9.10 Two nonzero polynomials $f(t,x)$ and $g(t,x)$ in two variables have only finitely many common zeros in ${\Bbb C}^2$, ...
3
votes
1answer
49 views

Galois cover an affine scheme

Let $X = \operatorname{Spec}(A)$ be an affine scheme, with $A$ noetherian (and normal if this is useful). We suppose that $X$ is a finite étale covering of $Y = \operatorname{Spec}(B)$, Galois with ...
3
votes
1answer
51 views

About two isomorphic schemes

My question is related to an answer I read on MO: http://mathoverflow.net/questions/157973/classical-algebraic-varieties-vs-k-schemes-vs-schemes In the accepted answer, the user Julian Rosen claims ...
3
votes
1answer
49 views

Why is $P$ singular

This is from Shafarevich's 'Basic Algebraic Geometry 1': Let $P=(\alpha,\beta)\in X$, and suppose that the equation of $X$ is written in the form $f(x,y)=a(x-\alpha)+b(x-\beta)+g$, where $g$ is a ...
3
votes
1answer
35 views

Curve with acnodes over closed fields?

From Wikipedia: An acnode is an isolated point not on a curve, but whose coordinates satisfy the equation of the curve. The term "isolated point" or "hermit point" is an equivalent term. I was ...
3
votes
1answer
75 views

Vanishing of higher direct images of a composition

In a paper I am studying we have the following situation. Let $S$ be the spectrum of a Dedekind domain, and let $X$, $Y$ and $Z$ be scheme of finite type over $S$, where $X$ and $Y$ are smooths and ...
3
votes
1answer
53 views

Coherent sheaves of finite length over $\mathbb{P}^n_k$

Let $k$ be an algebraically closed field. Are there any nonzero coherent sheaves on the projective space $\mathbb{P}^n_k$ that are supported at (only) finitely many closed points? If they don't exist, ...
3
votes
1answer
45 views

$K$-rational points and algebraic sets

Notation: If $X$ is a $K$-scheme, then a point $x\in X$ is said $K$-rational if its residue field $k(x)=\frac{\mathcal O_{X,x}}{\mathfrak m_{X,x}}$ is isomorphic to $K$. The set of all $K$-rational ...
3
votes
1answer
64 views

Pullback of an ample line bundle through a projection

Assume $X_1$ and $X_2$ are two smooth projective curves and let $M_i$ be an ample line bundle on $X_i$, for $i=1,2$. Further, denote the natural projection map on the $i$-th factor by ...
3
votes
1answer
80 views

$Z\subset \mathbb{P}^n$ irreducible iff its pre-image in $\mathbb{A}^{n+1}-\{0\}$ irreducible

I'm having trouble with this question (it's a homework question). If $p:\mathbb{A}^{n+1}-\{0\} \rightarrow \mathbb{P}^n$ is the canonical projection and $Z\subset \mathbb{P}^n$ is closed, then $Z$ is ...
3
votes
1answer
62 views

Is the universal hyperplane section the blowup of the baselocus?

I think I've heard this statement before but I'd like to make sure it's true. Let $X$ be a variety and $L$ a line bundle on it. Take $S < P\left(H^0(X,L)\right)$ to be a linear subspace of the ...
3
votes
1answer
93 views

Intuition on formal neighborhood in a scheme

Let $X$ be a Noetherian scheme, $x \in X$ a closed point. Denote by $\hat X$ the completion of $X$ along $x$. Now assume that two coherent modules $F, G$ on $X$ coincide over $\hat X$, i.e. $i^*F = ...
3
votes
2answers
93 views

Surjection from a Noetherian ring induces open map on spectra?

Let $A$ be a Noetherian ring, $f: A\rightarrow B$ a surjective ring map, then should the induced map on spectra $f^*: Spec(B)\rightarrow Spec(A)$ be an open map? In Atiyah and Macdonald, Chapter 1, ...
3
votes
1answer
68 views

Ampleness and global generation of divisors on smooth projective varieties.

Let $X$ be a smooth projective complex surface and $H$ an ample divisor on $X$. My main question is whether for any divisor $D$ we can say that eventually $mH+D$ will be globally generated. ...
3
votes
1answer
46 views

Unramification stable under change base

I want to show that if $f:X\to Y$ is an unramified scheme morphism (ie $m_y\mathcal{O}_{X,x}=m_x\mathcal{O}_{X,x}$ and $k(x)\leftarrow k(y)$ finite and separable) then any base change $X\times_Y Z\to ...