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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5
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3answers
119 views

Properties characterized by a vanishing Ext or Tor module

While reading Weibel's "An introduction to homological algebra'', I've noticed that many properties of a module are characterized by the vanishing of some Tor or Ext. Fix a (commutative) ring $R$ and ...
0
votes
1answer
27 views

three singular points

I am trying to prove the following: Let $F$ be homogeneous polynomial of degree $3$ in $k[X, Y, T ]$ where $k$ is algebrically closed field. Assume that $V(F)$ has three distinct singular ...
5
votes
1answer
56 views

What is the canonical bundle of a smooth divisor?

I am currently trying to learn some complex geometry, using mainly the book by Huybrechts. There is one thing confusing me, however: For example on Wikipedia people are talking about the canonical ...
1
vote
1answer
44 views

Are unipotent algebraic groups connected?

Is a unipotent algebraic group over a field of characteristic zero always connected?. As far as I know, every unipotent algebraic group over field of characteristic zero is isomorphic to a closed ...
4
votes
0answers
36 views

Generic cubic in $\mathbb{P}_{\mathbb C}^3$

I read that "two generic cubics are not projectively equivalents in $\mathbb P_{\mathbb C}^3$". I want to know if I understand well the statement. Does "generic cubics" means that all the cubics ...
2
votes
1answer
36 views

Closed and open subsets of Spec(R)

Suppose we are given a commutative ring with unit $R$ and an ideal $I \subseteq R$. I wondered if then the following is true: $$ V(I) \text{ is open in Spec}(R) \Leftrightarrow I = I^2. $$ I know that ...
0
votes
1answer
30 views

Calculating points of intersection and their multiplicities

p=(0,0), C: y=x^2, D: y=2x^2. Using Bezout's theorem and symmetries, show that ip(C,D)=2. Show this one more time using a parametrization of the conic. First off, Bezout's Theorem says there are 4 ...
0
votes
1answer
31 views

Determinant of this coherent sheaf on a surface $S$

If $C$ is a curve on a surface $S$, i.e. $i:C\subset S$, and $G$ is a line bundle on $C$, then $G|_U\cong \mathcal{O}_C$ where $U$ is an open subset of $S$, that is, $G$ is trivial on the complement ...
1
vote
1answer
27 views

Equivalent conditions for a closed immersion of schemes

In Hartshorne, a closed immersion of schemes is defined to be a scheme morphism $\Phi \colon Y \to X$ such that $\Phi$ is a homeomorphism onto $\Phi(Y)$, $\Phi(Y)$ is closed in $|X|$ and $$ (*) ...
1
vote
1answer
28 views

Blowup of smooth subscheme of smooth scheme is smooth

In Vakil's notes (http://math.stanford.edu/~vakil/216blog/FOAGjan2915public.pdf), Theorem 22.3.10, he shows that, if $X\hookrightarrow Y$ is a closed embedding of smooth varieties over $k$, then ${\rm ...
1
vote
1answer
81 views

Intersection multiplicity , Hilbert samuel polynomials and normal cones

I'm in the process of reading some parts of Fulton's "Introduction to intersection theory" and there's a short part there which I don't quite understand where I think I'm missing something obvious. ...
1
vote
2answers
58 views

Given 2 points where a line and a curve cross, find the third point where line and the curve cross.

The curve $y^2 = x^3 + 8$ contains the points $(1,-3)$ and $($$-7\over4$,$13\over8$$)$. The line through these two points intersects the curve in exactly one other point. Find this third point. P.S. ...
0
votes
0answers
12 views

Number of isotropic vectors of a hermitian form

Good evening, I have a question about isotropic vectors in hermitian spaces and I hope someone can help me out. Let K be a local non-dyadic field and $\pi$ a prime element (so 2 is a unit). Let $h$ ...
7
votes
2answers
83 views

Surfaces in $\mathbb P^3$ not containing any line

Let $d \geq 4$. I'm interested by know if there is a surface $S$ of degree $d$ in $\mathbb P^3_{\mathbb C}$ such that $S$ does not contains a line. I know I have no idea how to do it.
3
votes
1answer
52 views

When is the tensor product of a separable field extension with itself a domain?

I'm reading Algebraic Geometry and Arithmetic Curves by Qing Liu. On page 92, in the proof of Corollary 3.2.14 d), he states that if $K \otimes_k K$ is a domain, then $K = k$. Here $K$ is a separable ...
2
votes
0answers
68 views

Smoothness and algebraicity

My question is the following : given a complex algebraic affine variety $M \subset \mathbb{C}^n$, let us assume that it has moreover a smooth differential structure as a submaifold of ...
2
votes
1answer
34 views

Small question regarding gluing sheaves

Let $X$ be a topological space and $X = \cup U_i$ an open cover of $X$. Suppose we have sheaves $F_i$ on $U_i$ along with isomorphisms $\phi_{ij}: F_i |_{U_i \cap U_j} \rightarrow F_j |_{U_i \cap ...
0
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0answers
28 views

simple question about a translation

I've found this term "droit exceptionelle" readind an article by A.Beauville. He is talking about the blow-up of a point. He sais that $E$ is the droit exceptionnel apparue dans l'eclatement de $Q$. ...
0
votes
0answers
34 views

Line bundle of degree 1 on a genus 2 surface with 2 independent global holomorphic sections

By Riemann-Roch, for a degree 1 line bundle on a genus 2 Riemann surface the space of global holomorphic sections has dimension between $0$ and $2$. Is there an explicit example of a degree 1 line ...
3
votes
0answers
43 views

About the uniqueness of a quadric determined by sufficient points

What is the condition on a set of 14 points for them to uniquely determine a quadric in $\mathbb{P}^4$? Is being in linear general position enough to guarantee uniqueness? If not, what is the ...
2
votes
1answer
90 views

Complement of a hypersurface

I need your help in the following Lemma I.4.2 from Hartshorne: Let $Y$ be a hypersurface in $\mathbb A^n$ given by the equation $f(x_1,...,x_n)=0$. Then $\mathbb A^n\setminus Y$ is isomorphic to ...
1
vote
0answers
40 views

Question about Hartshorne Example 6.11.4 in Chapter II

I have a question about example 6.11.4 (page 142-143) of Hartshorne's Algebraic Geometry. In this example we are trying to determine the Cartier divisor class group of the cuspidal cubic curve $y^2 z ...
4
votes
1answer
63 views

Galois representations and isogenies of elliptic curves

Let $E$ be an elliptic curve over $\mathbb{Q}$. For each prime $\ell$, the action of $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on $E[\ell]$ (the group of $\ell$-division points of $E$) defines a ...
2
votes
1answer
67 views

Riemann-Roch theorem without heavy tools

I have read two proofs of Riemann-Roch : one very quick in Forster, Lecture on Riemann Surfaces which use cohomology of sheaf, and results from functional analysis. Another one is in the book of ...
1
vote
1answer
28 views

sufficient condition for varieties to meet transversally

Let $X$ be a smooth variety and $Y, Z$ a subvarieties of $X$. If $Y, Z$ are smooth and $dim(Y\cap Z)=dimY+dimZ-dimX$, then can I say that $Y$ and $Z$ meet trasversally? Are some more conditions ...
2
votes
0answers
109 views

If $p_{1},…,p_{r}\in\mathbb{P}^{n}$ are in general position, then $p_{1},…,p_{r-1}$ are in general position.

Let $p_{1},...,p_{r}\in\mathbb{P}^{n}=\mathbb{P}^{n}_{K}$. I have to prove that, if $p_{1},...,p_{r}$ are in general position , then $p_{1},...,p_{r-1}$ are in general position, but with a special ...
2
votes
0answers
44 views

zeta function of abelian varieties and the exterior algebra

Let $X$ be a smooth projective variety over a finite field $k$. By Grothendieck, the zeta function of $X$ admits the cohomological expression $$ Z(X, t)=\prod_{j=0}^{2\dim X} \det (1-F t \ | \ ...
1
vote
1answer
72 views

Conditions for $f_*(O_S)=O_B$

Let $S$ be a smooth projective surface, $B$ a smooth curve. Suppose $f\colon S\to B$ is a surjective morphism. Is there a condition for $f_*(O_S)=O_B$? (I read about a sufficient condition: the ...
2
votes
1answer
27 views

Composition of dominant rational maps

I have a question regarding the answer of the following question: how to define the composition of two dominant rational maps?. I'm sorry to open a new question for this, but I can't comment an answer ...
3
votes
0answers
48 views

Question about Proj of graded ring

Let $A=k[x_0, x_1,x_2]$ be a polynomial ring such that $\text{deg}(x_0)=2$, $\text{deg}(x_1)=\text{deg}(x_2)=1$. How can I understand what is $X=\text{Proj}\,A$?
3
votes
1answer
57 views

Cohomology of Segre varieties

Let $\Sigma_{n,m}$ be a Segre variety, i.e. the image of the Segre map $\mathbb{P}^n\times\mathbb{P}^m\to\mathbb{P}^{(n+1)(m+1)-1}$. Then how can I calculate the first cohomology group of its ...
1
vote
1answer
54 views

Looking for a proof that the resultant is the product of the differences of roots

I'm trying to find a general proof to an exercise given in Garrity et al's book, Algebraic Geometry: A problem-solving approach. The problem is this: Given two polynomials f and g, show that for ...
0
votes
1answer
55 views

A question about minimal rational surface

Let $S$ be a minimal rational surface. We can find a smooth rational curve $C$ with $C^2\ge 0$ [Complex Algebraic Surfaces, Beauville,p59]. Further more we assume that $C^2=m$ is minimal and $C.H$ ...
1
vote
0answers
40 views

derived versions of natural isomorphisms

I have just recently started approaching the topic of derived categories in algebraic geometry, and I'm doing so reading Huybrechts "Fourier-Mukai transforms in algebraic geometry". I have a doubt ...
0
votes
1answer
64 views

Finding maximal ideals and Krull dimension

I have difficulties in finding the maximal ideals and compute the dimension of a quotient ring $\mathbb{C}[x,y,z]/(x^2-y^2,z^2x-z^2y)$. Here $(x^2-y^2,z^2x-z^2y)$ is a product of $(x+y,z^2)$ and ...
1
vote
1answer
49 views

Evaluating rational functions

In Hartshorne's Algebraic Geometry the function field $K(Y)$ of a variety $Y$ is defined as the set of equivalence classes $<U,f>$ with $f$ being a regular function on the open subset ...
1
vote
1answer
35 views

Sanity check on the definition of a group action over an affine variety

On exercise 2.6.4 (page 32) of Andreas Gathmann's Algebraic Geometry class notes, you can read the following: "Let $X$ be an affine variety, and let $G$ be a finite group. Assume that $G$ acts on ...
0
votes
1answer
56 views

A simple question about rational Hodge conjecture

Good evening everyone : In the link here I found the following sentence : The Hodge conjecture predicts that the $\mathbb{Q}$ - linear span of the classes of algebraic subvarities in the cohomology ...
2
votes
1answer
54 views

Stable filtrations

An $\mathfrak{a}$-stable filtration is (as many know) an $\mathfrak{a}$-filtration $\{M_n\}$ such that for large $n$; $\mathfrak{a}M_n=M_{n+1}$. This is saying in some sort of way (I think) that the ...
0
votes
1answer
69 views

Parametric equation of a circle in 3 space - odd result

I have the following problem involving parameterized circular equations, but am getting strange answers and wanted to check if my approach made any sense. In 3D space, the parametric equation of a ...
2
votes
1answer
51 views

Hartshorne III 9.5 confused about base extension.

Hartshorne III Proposition 9.5 states: Let $f:X\to Y$ be a flat morphism of schemes of finite type over a field $k$. For any point $x\in X$ let $y=f(x)$. Then $$\dim_x(X_y)=\dim_x X-\dim_y Y$$ ...
1
vote
2answers
90 views

Dimension of local rings on scheme of finite type over a field.

In chapter III Hartshorne seems to be using without proof or mention a theorem on the dimensions of local rings of schemes of finite type over a field. I know that for an integral scheme of finite ...
1
vote
0answers
38 views

Morphism between projective schemes induced by injection of graded rings

Let $A$ be a graded ring and $d>0$ be an integer. Define the graded ring $B$ such that $B_i=A_i$ if $d$ divides $i$ and $B_i=0$ otherwise. Is it true that a homomorphism of graded rings ...
3
votes
1answer
38 views

Correspondence between Weil divisor and Cartier divisor on normal varieties

Let $X$ be normal singular projective variety over complex number, and $D$ be a Weil divisor on $X$. Is there an integer $n>0$ such that $nD$ is actually a Cariter divisor? If not, under what ...
0
votes
0answers
29 views

Ringed spaces isomorphism

Which of the following ringed spaces are isomorphic over $\mathbb{C}$? (a) $\mathbb{A}^1\backslash\{1\}$ (b) $V(x_1^2+x_2^2)\subset \mathbb{A}^2$ (c) $V(x_2-x_1^2, x_3-x_1^3)\backslash \{0\} ...
2
votes
2answers
44 views

Basic question related to stalk of the structure sheaf of a scheme

Let $X$ and $Y$ be schemes. Suppose I have a morphism of schemes $(\pi, \phi)$, where $\pi: X \rightarrow Y$, thus $\pi$ is continuous, and $\phi: O_Y \rightarrow \pi_*O_X$. Let $p \in X$ and $\pi(p) ...
2
votes
2answers
49 views

Quasi-Finite + Affine -> Finite?

If $f:X\to Y$ is an affine morphism of schemes, say with $Y$ irreducible, that is quasi-finite - all of the fibers, including the generic fiber, are finite - is it true that $f$ is finite? If not, ...
1
vote
1answer
64 views

Problem 2.26 in Fulton's Algebraic Curves: redundant hypothesis?

The problem reads: "Let $R$ and $S$ be DVRs with maximal ideals $M = (q)$ and $N = (p)$ respectively, $K$ the quotient field of $R$. Suppose $R \subset S \subset K$, and suppose that $M \subset N$. ...
1
vote
0answers
30 views

pull-back of canonical divisor under blow-up of a singular point

I was checking an example of canonical singularities from surface. We consider the surface $X:(xz=y^2)\subset \mathbb A^3$. The only singular point is the origin. We write down one affine piece of ...
0
votes
1answer
105 views

poles and zeros of function field of $\mathbb{P}^1$.

In which condition: an element of function field of $\mathbb{P}^1$ has zero or pole or no-zero&no-pole. I am thinking that: since $\mathbb{P}^1$ and $\mathbb{A}^1$ is birrationally equivalent ...