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)

2
votes
0answers
48 views

Lefschetz Hyperplane Theorem for Picard groups of surfaces?

Griffiths and Harris, On the Noether-Lefschetz Theorem and Some Remarks on Codimension-Two Cycles, Math. Ann. 271, 31-51 (1985), states [...] look at the restriction $$r_1 : ...
0
votes
0answers
26 views

Clarifying the definition of local equations of a subvariety

Let $X$ be an an algebraic variety over $\mathbb{k}$, $Y\subset X$ - its subvariety, $x\in Y$ - some point, $\mathcal{O}_x$ - its local ring. A family of functions $f_1,f_2,\ldots f_n$ is said to ...
2
votes
1answer
50 views

Varieties over a field $K$ are also varieties over any subfield of $K$.

Suppose that $f:X\longrightarrow\text{Spec} K$ is a variety over $K$, namely $X$ is an integral, separated $K$-scheme of finite type. Now if $L$ is a subfield of $K$, it is clear that there exists a ...
0
votes
0answers
21 views

Primality of homogeneous ideal

Let $R$ be the polynomial ring over the finite field $\mathbb{F}_p$ with $n$ variables. Let $I$ be an ideal of $R$ generated by homogeneous polynomials whose coefficients are 1 or -1. Are there any ...
1
vote
0answers
35 views

Isomorphism of stalks and the complement of the exceptional locus.

I am reading Qing Liu's book on Algebraic Geometry, and on pg. 272, the proof of lemma 2.20 b) there is a certain part I don't get. Let $X,Y$ be Noetherian integral schemes and let $f:X \rightarrow ...
1
vote
1answer
38 views

Under which assumptions counit of the adjunction $f^* f_* \to 1$ is epimorphic?

Let $f: X \to Y$ be a morphism of schemes. It produces a pair of adjoint functors $f^*$ and $f_*$ on the category of quasi-coherent sheaves i.e. there is a natural isomorpism $$ ...
0
votes
1answer
51 views

Quick question: a 2:1 map onto the projective line

Given a line $L$ in $\mathbb{P^2}$. How do we see that a surjective map $\mathcal{O}_\mathbb{P^2}^{\oplus2}\rightarrow j_{*}{\mathcal{O}_L(2)}$ ($j$ is the inclusion of $L$ to $\mathbb{P^2}$) ...
1
vote
1answer
34 views

Isomorphism of affine schemes

I have read that the morphism between affine schemes $(X,O_X)$ and $(Y, O_Y)$ is an isomorphism if and only if it induces an isomorphism of the global sections. I was interested in proving ...
2
votes
1answer
86 views

Prove that two definitions are equivalent

Preliminary notion: Suppose that $X$ is an algebraic variety over a field $K$ ($K$-scheme, integral separated, of finite type). If $L$ is a subfield of $K$, we say that $X$ is defined over $L$ if ...
0
votes
1answer
66 views

Hartshorne Chapter II exercise 5.7 on Invertible sheaves

I'm working on part c) which is to prove that for a Noetherian scheme $X$, a coherent sheaf $\mathscr{F}$ is invertible (locally free of rank 1) iff there exists a coherent sheaf $\mathscr{G}$ such ...
0
votes
1answer
37 views

Books which defines higher differentials in algebraic curves context

I'm reading an article which mentions a lot about higher differentials: I don't know what is $\Omega^n(F)$, my background is just Fulton's Algebraic curves book which defines just $\Omega(F)$. I ...
2
votes
0answers
39 views

Brauer groups of curves and base change

Let $X/k$ be a smooth, projective curve over $k$ and let $L/k$ be a finite extension of fields, where $k$ is a finite extension of $\mathbb{Q}_p$, $p \not=2$. Suppose $k(X)$ contains no elements ...
2
votes
2answers
130 views

Precise definition of affine, smooth, and irreducible

A book which I'm reading now says that "the Drinfeld curve $$ \mathbf{Y} = \{\, (x, y) \in \mathbf{A}^2(\mathbb{F}) \mid xy^q - yx^q = 1 \,\}$$ is affine, smooth, and irreducible." Here $p$ is an odd ...
0
votes
2answers
46 views

On affine scheme

Let $(X,O_X)$ be an affine scheme. By definition we know that $(X,O_X) \cong (Spec \ R , O_{Spec \ R})$ for some ring $R$. In fact we can show that we could take $R$ to be the global section ...
6
votes
1answer
119 views

Non-surjective but injective real polynomial functions $\mathbb{R}^n\to \mathbb{R}^n$

Over algebraically closed fields $K$, the Ax–Grothendieck theorem (see also this thread) states that injective polynomial functions $K^n \to K^n$ in $n$ variables are surjective. Is there a simple ...
2
votes
0answers
30 views

Doubt in a proof on projective varieties from Hartshorne

In Algebraic Geometry by Hartshorne in the proof of theorem 3.4 in Chapter 1 he gives an isomorphism of $k[y_1,...,y_n]$ with $k[x_0,...,x_n]_{(x_i)}$ by sending $f(y_1,...,y_n)$ to ...
2
votes
1answer
52 views

Why this is true using Riemann-Roch theorem

Let $C$ be a curve of genus $g$ over an algebraically closed field $k$ and $K=k(C)$ the field of rational functions of $C$. Consider $P$ a point at $C$. What I know: For each $r\in \mathbb N$, we ...
0
votes
2answers
41 views

Basic problem about property of algebraic set

We know there is a property about algebraic set: $V(I\cap J) = V(I) \cup V(J)$ Where $I$ and $J$ are two ideals consisting polynomials contained in a polynomial ring (not shown). $V(I)$ is a ...
1
vote
0answers
14 views

Proof of existence of a rational curve on a Fano variety with degree bounded, in Debarre's book

I am reading Debarre's book, Higher-Dimensional Algebraic Geometry. On page 61, we have the following theorem: if $X$ is a Fano variety of dimension $n > 0$, then through any point $x \in X$ there ...
3
votes
0answers
47 views

Pushing forward vector bundles on a plane curve via projection from a point

Let $C \subset \mathbb{P}^2$ be a smooth plane curve, $P \in \mathbb{P}^2$ is point not on $C$, consider projection from this point $$ \pi :\mathbb{P}^2 - \{P\} \to \mathbb{P}^1, $$ and restrict this ...
0
votes
1answer
27 views

The elements of the coordinate ring can not be regarded as functions (projective case)

I'm reading Fulton's algebraic curves and I have questions on page 46: 4 I know these fact are very basic, but I didn't understand why no elements of $\Gamma_h(V)$ can not be regarded as functions ...
5
votes
2answers
96 views

Principal $\mathrm{SL}_n$-bundles

It seems to be well-known that a principal $\mathrm{SL}_n$-bundle on a scheme or manifold $X$ is the same as a vector bundle of rank $n$ whose determinant is a trivial line bundle. One direction is ...
2
votes
1answer
41 views

Dimension of moduli of lines on quadric

What is the dimension of the moduli space of lines on a general quadric hypersurface in $\mathbb{P}^n$? Maybe the question is quite trivial, but different intuitive approaches (à la Italian algebraic ...
5
votes
0answers
43 views

How can a finite graph be viewed as a discrete analogue of a Riemann surface?

In the paper "Riemann–Roch and Abel–Jacobi theory on a finite graph" by Baker and Norine, the first line of the abstract states: "It is well known that a finite graph can be viewed, in many respects, ...
0
votes
0answers
47 views

The function field of $V=Z(y^2-x^3)$

Let $k$ be a field and let $V=Z(y^2-x^3).$ Can someone explain to me why $k(V)\cong k(s,t)$ ?? with $t=x+(y^2-x^3),s=y+(y^2-x^3)\in A(V)=k[x,y]/(y^2-x^3).$ Can we generalize it : If $V=Z(f)$ with ...
0
votes
1answer
21 views

maximal subtorus of a connected commutative algebraic linear group [closed]

I'm wondering the following: is the maximal subtorus of a connected commutative algebraic linear group over $k$ a) normal and closed b) defined over $k$ (for $k$ a field of characteristic zero, ...
6
votes
1answer
80 views

What are the points of some schemes?

Let $X=\operatorname{Spec}\mathbb{C}[x,y,t]/(xy-t)$, $Y=\operatorname{Spec}K[x,y]/(xy-t)\rightarrow \operatorname{Spec}K$ and $Z=\operatorname{Spec}R[x,y]/(xy-t)\rightarrow \operatorname{Spec}R$, ...
1
vote
0answers
30 views

Siegel's theorem and singular curves

I notice that often Siegel's theorem (there are only finitely many integral points on a curve of genus greater than 0) is stated with the requirement that the curve be smooth. Other times the ...
1
vote
0answers
51 views

How to show $\mathbb{A}_k^2 - \{ (x,y) \}$ is not an affine scheme

I am reading Ravi Vakil's notes and on page 137 (June 11, 2013 ver.) he explains why $U = \mathbb{A}_k^2 - \{ (x,y) \}$ is not an affine scheme in the following way (Please note I am paraphrasing it ...
0
votes
0answers
21 views

Special case of Bernstein theorem

There is a Bernstein theorem which gives an estimate on the number of complex non-zero roots of system of polynomial equations. Bernstein theorem. The number of solutions in $(\mathbb{C} \setminus ...
3
votes
2answers
68 views

Definition of intersection multiplicity in Hartshorne VS Fulton for plane curves

In exercise I.5.2 in Harshorne there is the following definition of intersection multiplicity for two curves in $\mathbb{A}^2$: \begin{equation} \mathrm{length}_{\mathcal{O}_P}\mathcal{O}_P / (f,g) ...
0
votes
0answers
47 views

What is the meaning of this notation in algebraic geometry (from /): $k\left[x_{1},\ldots,x_{r}\right]\mathbf{/\left(f_{1},\ldots,f_{r}\right)}$?

I have stumbled on something is apparently a trivial concept, but the difficulty is that I haven't seen this notation before. Here is the fragment of a text from lecture notes: Let us call $\rho$ ...
0
votes
0answers
46 views

Visualizing Line Bundles of Projective space

How should I visualize $\mathscr{O}_{{\mathbb{P}_{\mathbb{C}}}^1}(n)$?
1
vote
0answers
27 views

EGA reference for equivalent criteria for ampleness

Let $X$ be a projective scheme over a field $k$ with $\mathcal{L}$ a very ample line bundle on $X$ (very ample here means relative to the structure morphism $X \to \operatorname{Spec} k$. Where is it ...
0
votes
1answer
33 views

Quotientes of afine group schemes

Let $G=Spec(K[x_1,...,x_n])$ an afine group scheme and $H$ an subgroup scheme of $G$ then -Can i say that $H$ is a afine group scheme, if not then when can I say it? -How can I define the quotient ...
1
vote
1answer
20 views

Evaluating a function on a locally ringed space

I am just learning about locally ringed space. Let $(X, O_X)$ be a locally ringed space. It (the notes I am reading) says that: The $O_X,p$ is a local ring for each $p \in X$. Let $m_p$ be the ...
2
votes
1answer
32 views

Conic through 4 points

Let $p_1,\ p_2,\ p_3,\ p_4$ be any 4 different points on $\mathbb{CP}^1$ and $x_1,\ x_2,\ x_3,\ x_4$ are 4 different points on $\mathbb{CP}^2$. How can I show that there is unique conic $Q$ passing ...
1
vote
0answers
18 views

extension maps moduli space

I know that given $\phi: X \to \mathbb{P}^n$ a rational map, where $X$ is for example a projective curve, $\phi$ can always be resolved as a sequence of blowups. Now I consider the map $\phi$ that ...
1
vote
0answers
70 views

What are the “hidden” symmetries in Goldbach Conjecture?

What are the "hidden" symmetries in Goldbach Conjecture ? If Goldback conjecture is true, the basic instinct is that there must exist some "symmetries" which ensure (and lead) such properties. As we ...
2
votes
1answer
27 views

Problem about the kodaira's dimension

I have this exercise: Let $S$ a surface equal to the cartesian product of two projective irreducible smooth curves of genus greater or equal than 1. So $S=E \times F$. I want to describe the ...
2
votes
0answers
52 views

What are the missing gaps to prove Goldbach Conjecture?

When Andrew Wiles proved FLT, all he needed to do was to prove "semi-stable elliptic curve case" of Shimura-Taniyama conjecture. He did not need to start from scratch, he just needed to fill this ...
0
votes
0answers
21 views

what should do to start in dimension theory

I want to start reading something in dimension theory specially the defenition of a dimension of a ring and a connection between dimA and dimA[x] and depth of the prime ideal and... would you please ...
0
votes
1answer
20 views

The normalized valuation on $\mathcal{O}_{P,Y}$

Let $Y$ be a curve, and $P\in Y$ a smooth point. So $\mathcal{O}_{P,Y}$ is a regular local ring of dimension one, therefore is a discrete valuation rings. My question is what is the normalized ...
0
votes
1answer
39 views

Notation in Hartshorne book about monoidal transformations (blow-ups of surfaces along points)

I'm starting the process of learning the concept of blow-up for surfaces along a point. At page 386 of Hartshorne's book, the author defines the monoidal transformation of a surface $X$. But at the ...
0
votes
0answers
37 views

Question about Corollary I.6.6 in Hartshorne

I am having trouble understanding something in Corollary I.6.6 of Hartshorne. Let $K$ be a function field of dimension one over $k$ (by which he means a finitely generated extension of transcendence ...
3
votes
1answer
46 views

Some questions on the formation of the BSD conjecture

I'm quite curious how Birch and Swinnerton-Dyer formed their famous conjecture in the beginning of 1960s. I read some paper of Birch and Swinnerton-Dyer, as well as some paper of Tate and several ...
0
votes
0answers
27 views

is there an existing formula in finding the area of a rhombus wherein only the side is given?

is there an existing formula in finding the area of a rhombus wherein only the side is given? No measure of angles, no lengths of diagonals , height, etc. is given.
0
votes
1answer
45 views

Why is the multiplicative subgroup of a field an affine algebraic group?

Let $K$ be an algebraically closed field. Let $G_m$ be the multiplicative subgroup of $K$. In Lectures on Linear Algebraic Groups by Tamás Szamuely (you can easily find these notes online) it is said ...
5
votes
1answer
83 views

Bounding the cohomology of a smooth projective variety

Let $X/\mathbb C$ be a smooth projective variety. Suppose it is smoothly embedded in $\mathbf P^n$ as the zero locus of an ideal generated by homogeneous polynomials $f_1, f_2, \dots, f_r$ in $n+1$ ...
0
votes
0answers
31 views

image of the canonical morphism of a spanned divisor

Take $S$ a complex algebraic projective surface and $M\in Div(S)$ a spanned divisor of $S$. Due to the fact that $M$ is spanned we can define a morphism $\phi_{[M]} :S \rightarrow \mathbb{P}^N$ with ...