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)

1
vote
0answers
25 views

group scheme of prime order p is killed by p

In the article "Group Schemes of Prime Order" by Tate and Oort (see here) it is proved that a group scheme of prime order $p$ over the base $S$ is killed by $p$ (Theorem 1). The authors state that it ...
0
votes
0answers
13 views

Rank 1 Azumaya algebra

Let $X$ be a locally Noetherian scheme. Let the topology on $X$ be etale. Let $A$ be an Azumaya algebra over the scheme $X$ of finite rank. It can be thought of as an element of the Brauer group ...
2
votes
1answer
51 views

Show that the meromorphic differential of the homogeneous polynomial is holomorphic and not isomorphic to $\mathbb{P_1}$

Consider the elliptic curve i.e. non-singular cubic, $X$ given by the equation $\xi_0\xi_2^2=\xi_1^3-\xi_0^2\xi_1$ in projective coordinates $(\xi_0:\xi_1:\xi_2)$, or, equivalently, by the equation ...
2
votes
1answer
65 views

Divisor of the meromorphic differential $\omega=\frac{dx}{y^3}$ on C: $\xi_1^4+\xi_2^4=\xi_0^4$

Consider Fermat's curve of degree 4 defined by C : $\xi_1^4+\xi_2^4=\xi_0^4$ in projective coordinates $(\xi_0 :\xi_1 :\xi_2)$ or, equivalently, by the affine equation $x^4 + y^4 = 1$ in the affine ...
1
vote
0answers
66 views

Analogue of locally constant sheaf in algebraic geometry

If I take just the definition of locally constant sheaves for algebraic varieties I get something pretty trivial (basically due to irreducibility); so how can one make a set up similar to what happen ...
1
vote
0answers
42 views

irreducible subvariety of a torus

Let $H$ be an irreducible subvariety which is also a closed subgroup of $(\mathbb{C}^*)^n=spm(k[x_1,x_2,...,x_n,y_1,...,y_n]/(x_iy_i-1,i=1,2...,n))$.How to show that $H$ is also isomorphic to a torus? ...
0
votes
1answer
30 views

Conditions of $f=a+bx+cz+dx^2+exz+fz^2+…$ such that its tangent line is $z=0$ and inflection point is at the origin.

Let $x,z$ be coordinates on $k^2$ and $f\in k[x,z]$; write $f$ as $$f=a+bx+cz+dx^2+exz+fz^2+...$$ Write down the conditions in terms of $a,b,c,...$ such that (a) $P=(0,0)\in C: (f=0)$; (b) the ...
1
vote
1answer
41 views

Direct limit sheaf.

Let $\{ \cal{F}_i, \mu_{ij}\}$ a direct system of sheaves and morphisms on a topological space $X$. Define the direct limit os the system $\{ \cal{F}_i, \mu_{ij}\}$ as the sheaf associated to the ...
0
votes
0answers
28 views

global section of some divisor

Let $f:X\longrightarrow \mathbb{P}^2$ be the blow-up at $p=(1,0,0)$ and let $D:(x_0x_1x_2=0)$. Set $D'=$ strict transform of $D$. Then how can I compute $h^0(X,\mathcal{O}_X(n(K_X+D')))$ for any ...
1
vote
0answers
21 views

Analytically determining whether a laser beam will hit a moving target

I'm tinkering on a space-related computer game. The objects of the game are in 3D space and their motions are defined by 3 3D vectors: ${vector}\ V: \{X, Y, Z\} \\ {motion}\ M: \{V_{position}, ...
0
votes
1answer
29 views

$X_1,X_2$ disjoint closed in $Spec(R)$ properties

This is a problem in three parts, I managed to prove the first part, but the others I couldn't. Let $R$ be a ring and let $X_1,X_2\subset Spec(R)$ be closed (in Zariski topology) and disjoint such ...
2
votes
1answer
73 views

Chern class of complex vector bundles

Let $\xi$ be an $n$-dimensional complex vector bundle. It is claimed that the Chern class of $\xi$ is $$ c(\xi)=(1+x_1)\cdots (1+x_n), $$ $|x_k|=2$, $c_j(\xi)$ is the $j$-th symmetric polynomial of ...
7
votes
2answers
90 views

How can hypersurfaces “know” the degree of their defining polynomials?

I'm currently trying to learn some complex and projective geometry. There is one issue bugging me again and again, from different perspectives, and I just can't get my head around it. One incarnation ...
0
votes
0answers
33 views

Finite union of algebraic affine varieties

I'm studying Commutative algebra and Algebraic Geometry. I have proved the following proposition: $V(I)∪V(J)=V(IJ)=V(I∩J)$ I would know why sometimes is better to use the product of ideals instead ...
1
vote
0answers
30 views

Relative line bundle along divisor $D$

Let $X$ and $B$ be a compact Kahler manifolds and $\pi:X\to B$ be a holomorphic surjective map and $D$ be a divisor in $X$ how can we define relative canonical line bundle on $B$ along a divisor $D$? ...
2
votes
2answers
64 views

$\sqrt{\frac{h^{2}}{4}+d^{2}-h d \cos x}-\sqrt{\frac{h^{2}}{4}+d^{2}+h d \cos x}$ equals $h\cos x$?

Trying to simplify the expression, I observed: $y=\sqrt{\frac{h^{2}}{4}+d^{2}-h d \cos x}-\sqrt{\frac{h^{2}}{4}+d^{2}+h d \cos x}$ graphically equals $y=h\cos x$ when pluging in arbitrary values of ...
2
votes
1answer
49 views

Can we recover étale, fpqc etc. morphisms of schemes from the affine versions?

$\DeclareMathOperator{\Spec}{Spec}$ Consider the following procedure for defining a class of morphisms of schemes: Take a suitable class of homomorphisms of rings (e.g., canonical maps to ...
0
votes
0answers
32 views

Extending a morphism

Let $k$ be a field and $X:=V(T_3^2-T_1T_2)\subseteq \mathbb{A}_k^3$. Let $Z:=\{(0,0,0)\}\subseteq X$ viewed as a closed subscheme with reduced scheme structure. Denote by $\widetilde{X}:=Bl_Z X$ the ...
4
votes
2answers
61 views

Representations of a quiver and sheaves on P^1

We know from Beilinson that there's an equivalence of derived categories $D^b Rep(Q) \simeq D^b Coh(\mathbb{P}^1)$ where the lefthandside is the derived category of bounded complexes of ...
1
vote
0answers
33 views

noether normalization and complete intersection

Let I be an ideal of the polynomial ring $P=K[x_{1},...,x_{n}]$ that is generated by degree two polynomials ${f_1,...,f_k}$. The zero set $\mathcal{Z}(I)$ is isomorphic to an affine space of ...
1
vote
1answer
40 views

Exercise in R.Vakil 18.4.L: Ample line bundles and finite morphism

Here's the question: R.Vakil, Exercise 18.4.L: Suppose $\mathcal{L}$ is a base-point free invertible sheaf on a proper variety $X$, and hence induces some morphism $\phi: ...
1
vote
1answer
42 views

Is the set of fixed points an algebraic variety?

Let $V$ be a finite dimensional $\mathbb{C}-$vector space. The linear action of its automorphism group $GL(V)$ on $V$ induces an action on the projective space $\mathbb{P}(V)$, i.e. $$ GL(V) \times ...
2
votes
1answer
30 views

How to classify quadratic forms using their signature

I just did a question asking to classify the kind of curve of a given quadratic polynomial: $$0=3x^2+8xy+6y^2+12x+20y+17$$ I completed the square a few times and eventually (correctly) observed that ...
2
votes
1answer
17 views

Find if two points are on the same sheets of a quadric surface

I have the equation of a hyperboloid cylinder (having two sheets), and I want to test if two points are on the same sheet or different sheet. How can I do this?
1
vote
1answer
25 views

Localization of R [duplicate]

Let p be prime ideal in the ring R. I want to show that there is a bijection between the set of prime ideals of R and the set of prime ideals of localization of R. I am quite confused about the ...
0
votes
1answer
60 views

example of proper ideal of C[x,y]

I am stuck in this problem for a while, and the main idea will be important for some exercises, so I really want to know how to find an example like this I need an example of an proper ideal, ...
1
vote
0answers
26 views

Algebraic extension and integral element

I'm doing the exercise of Algebraic curves by Fulton and I'm stucked in this problem Let $R$ be a domain with quotient field $K$, and let $L$ be a finite algebraic extension of $K$. (a) For any ...
4
votes
1answer
52 views

Exercise 2.12 in Harris - Algebraic Geoemetry: a first course

Consider the three lines of $\mathbb{P}^3$ given by $L: \, z_0 = z_1 = 0 \\ M: \, z_2=z_3 = 0 \\ N: \, z_0 = z_2, \, z_1 = z_3.$ It is claimed in Exercise 2.12 of Harris (a first course) that the ...
0
votes
0answers
14 views

$h$ invariant of forms and its relation to Birch rank

Let $k$ be a field. Given a form $F_i \in k[x_1,..., x_n]$ of degree $d$, the $h$-invariant $h_k(F_i)$ is defined to be the least positive integer $h$ such that $F_i$ can be written as $$ A_1B_1 + ...
2
votes
1answer
37 views

canonical representation of three skew lines in $\mathbb{P}^3$

Consider three skew (non-intersecting) lines $L,M,N$ in $\mathbb{P}^3$. Each line is given by two equations of the form $\alpha_{i,j}^\top z = 0, \, i=1,2,3, j = 1,2$, where $z=(z_0,z_1,z_2,z_3)$ are ...
4
votes
1answer
49 views

An isomorphism theorem for sheaves.

Let $\varphi: \cal{F} \longrightarrow \cal{G}$ a morphism of sheaves. My goal is to prove that $im\varphi \simeq \cal{F} / Ker \varphi$. My thoughts about this problem: 1) $im \varphi(U) \simeq ...
1
vote
1answer
61 views

Terminology: Why is it called sheaf COhomology?

I just learned the definition of sheaf cohomology as the derived functors of the global sections functor. I have a question about terminology - why is it called sheaf COhomology and not just homology ...
0
votes
0answers
34 views

Is Spec R compact? [duplicate]

And if so, why? I'm having some trouble with this. I know that the $D_{f}$ (set of primes not containing $f$) are the open sets and form a basis for the Zariski topology; but I do not know how to go ...
0
votes
0answers
26 views

Sheafification preserves injectivity.

I want to prove the following: Let $\varphi: \cal{F} \longrightarrow\cal{G}$ a morphism of presheaves such that $\varphi (U): \cal{F}(U) \longrightarrow\cal{G}(U)$ is injective for each $U$. Then the ...
2
votes
0answers
36 views

Is there error in proof of lemma on Riemann-Roch space of divisor $D$?

I'm reading Steven Galbraith "Mathematics of Public Key Cryptography" and can't understand lemma 8.4.2 on page 154 that ...
0
votes
0answers
14 views

The “good” singularities of a local model?

In the theory of Shimura Varieties you want to construct a model over the ring of integers of the reflex field of the Shimura variety. You want it to be flat and have "good" singularities. This ...
3
votes
1answer
50 views

Show that $(Y^2-X^3)|f$ if $f$ vanishes on the curve $C: (t^2,t^3)$, and determine what property of a field $k$ will ensure that the result holds.

Let $\phi: \mathbb{R^1}\rightarrow \mathbb{R^2}$ be the map given by $t \mapsto (t^2,t^3)$; prove directly that any polynomial $f\in \mathbb{R[X,Y]}$ vanishing on the image $C=\phi(\mathbb{R^1})$ is ...
0
votes
1answer
48 views

Picard Group Can Contain rational curve?

X is a smooth projective curve with genus>2 My Question is, Pic(X) can contain rational curve(P^1) or not
0
votes
0answers
26 views

Why is the scheme associated to a product of quasi-projective varieties naturally isomorphic to the product of the associated schemes?

What I mean by this is, suppose $X$ and $Y$ are quasi-projective varieties over some arbitrary field $k$. Then $X\times Y$ is again a quasi-projective variety. I've seen this a few times, but what is ...
1
vote
0answers
61 views

Hartshorne notation in section III.12

I am reading section III.12 in Hartshorne, the one about the Semicontinuity Theorem. For $f:X \rightarrow Y$, where $Y=\mathrm{Spec}A$ and $\mathcal{F}$ a coherent sheaf on $X$, he writes ...
1
vote
0answers
35 views

If $X$ is quasi-projective but the scheme $\tilde{X}$ is affine, is $X$ necessarily affine?

I'm curious if the following works as a criterion to determine when a quasi-projective variety is actually affine. If $X$ is a quasi-projective variety, and the scheme $\tilde{X}$ is affine, does ...
1
vote
0answers
18 views

A base point free linear system of conics in $\mathbb{P}^2$ giving a rise to a regular map

Let $X=\mathbb{P}^2$ and $C=V(x^2+y^2-z^2)$ an irreducible conic. Then the Riemann-Roch space associated to $C$ by definition is $\mathcal{L}(C)=\{ f \in k(X)^{*} \vert div f + C \geq 0 \} \cup \{0 ...
3
votes
1answer
78 views

Affine variety and dimension

I'm working on a paper about representation of quivers and Gabriel's theorems. See this .pdf if you're interested ; but I guess you can answer my question without knowing anything about quivers, or at ...
0
votes
1answer
26 views

Find all $b$ such that $(t,t+b)$ is tangent to circle.

This is problem 3 on page 147 of ideals varieties and algorithms. We use the following definition of tangent line: where multiplicity is defined as Statement of problem: Consider the straight ...
0
votes
0answers
38 views

sheafification and quasi-coheret sheaf

My problem comes from this post: http://math.stackexchange.com/a/467252/115619 The example he gave is $j_!\mathcal O_U$. But I think $j_!\mathcal O_U$ is the sheafification of the presheaf $V \to ...
1
vote
0answers
34 views

Elliptic Curve Group and Multiplicative Inverse of an element.

Suppose $E$ be an Elliptic Curve over a field $F_q$ and $q=p^n$ where $p=$ prime. We know that the Elliptic Curve group $E(F_q)$ under addition is an Abelian/Commutative Group of order, ...
0
votes
0answers
27 views

induced isomorphism on the blow up

Let $k$ be algebraic closed field and let $\mathbb{P}^2$ the projective space over $k$ of dimension 2. Consider the birational map $$f:\mathbb{P}^2 ---> \mathbb{P}^2, [x_0, x_1, x_2] \mapsto ...
5
votes
1answer
74 views

Flatness under reduction

Suppose that $f : X \to Y$ is a flat morphism of schemes. Is $f_\text{red} : X_\text{red} \to Y_\text{red}$ necessarily flat? Are there any hypotheses that would guarantee this?
0
votes
1answer
30 views

Working with an Affine Variety and Maps

We have a $\phi :\mathbb C^4 \rightarrow\mathbb C^4$ $$\phi (a_1,a_2:b_1,b_2) = \left(\begin{array}{cc} a_1b_1 & a_1b_2 \\ a_2b_1 & a_2b_2 \end{array}\right) $$ We want to argue there exists ...
1
vote
1answer
33 views

morphisms of projective varieties and induced ring homomorphisms

If $\phi: X \rightarrow Y$ is a morphism of affine varieties, then we get an induced homomorphism $\tilde{\phi} : A(Y) \rightarrow A(X)$ of their affine coordinate rings (and vice versa). Question ...