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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2
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0answers
43 views

Blowup of six points in $\mathbb{P}^2$

I have been looking at sequential blowups of six points $P_i$ in $\mathbb{P}^2$. In the general case, we can identify the blowup with a nonsingular cubic surface in $\mathbb{P}^3$ and under this ...
12
votes
0answers
153 views

Maximum number of intersection points of two different Bernoulli lemniscates

What is the maximum number of intersection points of two different Bernoulli lemniscates in the real plane? (Of course two identical lemniscates share an infinite number of points.) Here are ...
3
votes
1answer
42 views

Closed points are dense in $\operatorname{Spec} A$

From 3.6.J in Vakil: Let $k$ be a field, and let $A$ be a finitely generated $k$-algebra. We want to show the closed points are dense in $\operatorname{Spec} A$. This is the set of prime ideals of ...
3
votes
0answers
50 views

Assume that for any pair of vertices $P_i$ and $P_j$ , there exists a vertex $P_k$ of the polygon such that $∠P_i P_k P_j = \pi/3.$

Let $P_1 P_2 \dots P_n$ be a convex polygon in the plane. Assume that for any pair of vertices $P_i$ and $P_j$ , there exists a vertex $P_k$ of the polygon such that $∠P_i P_k P_j = \pi/3.$ Show ...
0
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1answer
44 views

Why does $\bigwedge^p(L_1\oplus\cdots\oplus L_p)\cong L_1\otimes\cdots\otimes L_p$?

I have a very brief question. If you have a bunch of line bundles $L_1,\dots,L_p$ over a scheme $S$, why does $\bigwedge^p(L_1\oplus\cdots\oplus L_p)\cong L_1\otimes\cdots\otimes L_p$, and can't find ...
3
votes
2answers
145 views

How is the notion of adjunction of two functors usefull?

Is there a secret or an intuitive idea behind the fact of creating the concept of adjunction of two functors ( Functor - Adjoint Functor ) ? How is this notion of adjunction of two functors usefull ? ...
5
votes
1answer
39 views

Write $\mathbb{P}^3_{\mathbb{C}}$ as a union of disjoint lines

Is there a set $\Gamma=\{L \subseteq \mathbb{P}^3_{\mathbb{C}}: L \textrm{ is a projective line}\}$ such that every point $p \in \mathbb{P}^3_{\mathbb{C}}$ lies on exactly one line $L_p \in \Gamma$? ...
2
votes
0answers
34 views

Union of affine varieities is a projective variety?

Let $X \subset \mathbb{P}^n$ be a subset and let $U_i = \{ [z_0: \cdots :z_n] : z_i \neq 0 \}$ for $0 \leq i \leq n$ be the usual affine cover of projective space. Suppose that $X \cap U_i$ is an ...
1
vote
2answers
103 views

Cohen-Macaulay rings and Normal rings

is there an example that R is Cohen-Macaulay but not normal ring? what about the converse example?
0
votes
1answer
24 views

What is vectors straddle a plane mean?

There is a condition in a paper, saying that two vectors straddle a plane. How can we transfer this condition to a equation? Because I have another 5 equations and need this one to solve 6 unknowns. ...
-1
votes
1answer
54 views

Why is this map injective?

On the page we find the following: $ \phi : Z ( P_1 , \dots , P_r ) \to \mathrm {Spm} (K[ X_1 , \dots , k_n ] / \sqrt{ ( P_1 , \dots , P_r )}) $ defined by $ \phi ( ( a_1 , \dots , a_n ) = \pi ( ( ...
1
vote
1answer
41 views

Number of inflection points of an algebraic projective curve

I' m trying to prove that a curve in $\mathbb{P^{2}(\mathbb{C})}$ of degree $d$ has an infinite number of inflection points or it has at most $\le 3d(d-2)$ inflection points. Let be $C$ the curve and ...
0
votes
0answers
22 views

Reflexive sheaves on stable curves-II

This is an extension of Reflexive sheaves on stable curves. Let $C$ be a stable curve and $\mathcal{F}$ a reflexive sheaf on $C$ supported on the whole of $C$. Is the projective dimension of ...
0
votes
1answer
29 views

Reflexive sheaf on normal surfaces

Let $X$ be a normal, projective scheme of pure dimension $2$ and $\mathcal{F}$ is a reflexive coherent sheaf on $X$. Is $\mathcal{F}$ locally free?
2
votes
1answer
49 views

Blow-up toric varieties.

I have to take a talk of an hour and I have to talk about blow-up of toric varieties. Can you suggest me some interesting examples that I can present? How can I find a good reference for the theory ...
1
vote
1answer
35 views

How does an irreducible quadric in projective space look like?

I read the answer to the following question: Quadrics are birational to projective space Here it is stated that: Over a field $k$ of characteristic ≠2 every irreducible quadric $Q \subset \mathbf ...
0
votes
2answers
43 views

Jacobian of n linearly independent forms in n variables

Let $k$ be a field of characteristic zero and let $f_1, \ldots, f_n \in k[x_1, \ldots, x_n]_d$ be linearly independent forms of degree $d$ in $n$ variables. Is there a nice algebraic argument for ...
0
votes
0answers
29 views

How to understand the elementary modification?

In the Friedman's book: Algebraic Surfaces and Holomorphic Vector Bundle , there is concept elementary modification at Ch2 Def15. Let V is rank 2 bundle on X and L a line bundle on effective divisor ...
1
vote
0answers
14 views

Automorphisms of del Pezzo surface of degree $1$.

I have som problems with understanding of finite subgroups $G$ of $Aut(S)$,where $S$ del Pezzo surface of degree $1$. I consider case, when $k = \mathbb{Q}$. I don't understand why $Aut(S)$ embedding ...
2
votes
0answers
27 views

Lifting of Commuting Maps of Vector Bundles

Assume that we have a vector bundle $\mathcal{F}$ over $\mathbb{P}^d(\mathbb{C})$ that is generated by global sections. Let $\pi \colon \mathcal{O}^n \to \mathcal{F}$ be the associated map that is ...
5
votes
0answers
65 views

Does $L_1\oplus\mathbb{A}^1_X\cong L_2\oplus\mathbb{A}^1_X$ imply that $L_1\cong L_2$?

Suppose $L_1,L_2$ are line bundles over a scheme $X$. If one knows that $L_1\oplus\mathbb{A}^1_X$ and $L_2\oplus\mathbb{A}^1_X$ are isomorhpic, is that enough to conclude that $L_1$ and $L_2$ are ...
-3
votes
1answer
46 views

Intersection of a hypersurface with a projective variety [closed]

I don't understand the argument in the proof of Corollary 3.15 (This is from Harris). In particular, how exactly is Corollary 3.14 applied?
2
votes
1answer
37 views

Sections of Divisors on Projective Space

Everything is over $\Bbb{C}$. Let $X$ be a smooth projective variety. Fix an open covering $U_i$ of $X$ and let $D$ be a Cartier divisor given by a collection of rational functions ...
3
votes
1answer
31 views

Are points in general position generic points?

In Harris' algebraic geometry book, $p_{1},\ldots,p_{r}\in\mathbb{P}^{n}$ are said to be in general position if no $n+1$ or fewer of them are dependent. I want to prove that, if ...
0
votes
0answers
16 views

Quick question: Pull back under double cover of tangent space on the projective plane is stable?

Let $f:\mathbb{P}^1\times\mathbb{P}^1\rightarrow\mathbb{P}^2$ be the double cover branched along some conic $C\subset \mathbb{P}^2$. Is $f^*T_{\mathbb{P}^2}(-1)$ $\mu$-stable/semistable? Is there any ...
1
vote
1answer
57 views

How do we get this quotient $\textrm{Ext}^1(N,M)/\textrm{Hom}(N,M)$?

If $0\longrightarrow M\longrightarrow E\longrightarrow N\longrightarrow 0$ is a short exact sequence of vector bundles on a surface. Here $M$ and $N$ are line bundles, and so rank $ E$=2. Also, if ...
1
vote
1answer
29 views

Map between free sheaf modules not arising from a “matrix”

My question is about this example in the Stacks project. For convenience and completeness, here is the relevant part. Let $X$ be countably many copies $L_1, L_2, L_3, \ldots$ of the real line ...
0
votes
2answers
47 views

How do I find an isomorphism between varieties

Our book defines an isomorphism between varieties when there exist two maps say $\phi: V \rightarrow W$ and $\psi: W \rightarrow V$ both morphisms and $\psi \circ \phi =id_V$ and $\phi \circ \psi =id ...
1
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2answers
72 views

What does a polynomial look like under projection of underlying space?

Consider a multivariate polynomial in $F:\Bbb R^3\rightarrow\Bbb R$, $F\in\Bbb R[x,y,z]$ with prescribed values over a sphere in $\Bbb R^3$. Consider standard Riemann projection from $\Bbb ...
4
votes
0answers
41 views

Global functions functor for derived stacks

On page 24, 25 of the paper Loop Spaces and Connections the authors refer to a functor $\mathcal{O}: DSt_k \rightarrow DGA_k^{op}$ from derived stacks to dg algebras over $k$. It is defined as ...
0
votes
1answer
14 views

Correspondence between morphism and ring of regular functions

in Hartshorne it is explained that an morphism of varieties $\varphi:X \to Y$ gives rise to $k$-algebra-homomorphism of $O(Y) \to O(X)$. Now I know by the defining property of morphism that a morphism ...
1
vote
1answer
35 views

Every variety contains open affine normal subvariety

How to prove this? I think that the starting point here is to use the fact that the set $\{x\in X \,|\, X\, \text{is normal at}\, x\}$ is open. What do I do next? Thank you in advance.
2
votes
1answer
61 views

Build sheaf from stalks

If I have a topological space $T$ and for each $p \in T$ I have an object $A_p$ in some category $\mathscr{A}$, then how can I define a sheaf out of this? In other words can I build a sheaf with ...
2
votes
0answers
98 views

How to prove that an ideal can not be generated by 2 elements

In Kunz's "Introduction to commutative algebra and algebraic geometry", page 137-139, particular monomial affine curves are described. Here is the link. In case the curve is not an ideal ...
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vote
0answers
30 views

Generalization of Euler theorem for homogeneous polynomials

Euler's theorem for homogeneous polynomials is well known. If $F:\mathbb{R}^{n}\rightarrow \mathbb{R}$ is a homogeneous polynomial, then we have: $x_{1}\frac{\partial F }{\partial x_{1}} + ... + ...
4
votes
0answers
55 views

Is the preimage of the non-normal locus a divisor?

Let $X$ be a complex, affine variety. Let $\nu:\tilde X\to X$ be the normalization of $X$ and denote by $D\subseteq X$ the closed set of points where $\nu$ fails to be an isomorphism, i.e. $D$ is the ...
1
vote
0answers
33 views

quick question about a definition of homeomorphism set classes

Take $S$ a surface of general type. I want to define $Q$ the set of homeomorphism classes determined by the surface $S$. How can i define $Q$? I think that $Q$ is determined by all surfaces $S^{'}$ ...
3
votes
1answer
57 views

Difference between Euler characteristics of a Riemann surfaces

Let $X$ be a compact connected Riemann surface of genus $g$. Let $U$ be the complement of $r$ points in $X$. The Euler characteristic of $X$ = $2-2g$. That I understand. But I'm confused about the ...
0
votes
1answer
15 views

“cover the unit sphere by c-fine grid” to prove the vector length preserved by random projection?

The below figure is extracted from the paper http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4031351 . I did not understand the techniques used in the proof, namely, 1."cover the unit ...
3
votes
1answer
60 views

Maximal ideal in a polynomial ring over a field that is not algebraically closed

I want to prove that although $K$ is a field that IS NOT algebraically closed, every maximal ideal in $K[x_1, \ldots, x_n]$ can be generated by $n$ elements. To prove this, I am following the next ...
0
votes
0answers
23 views

Is the profinite completion of $SL_2(\widehat{\mathbb{Z}})$ isomorphic to $\widehat{SL_2(\mathbb{Z})}$?

Is the profinite completion of $SL_2(\widehat{\mathbb{Z}})$ isomorphic to $\widehat{SL_2(\mathbb{Z})}$ ? Does anything change if we replace $SL$ with $GL$?
3
votes
0answers
53 views

degrees of L-functions and dimensions of Shimura Varieties

I try to grab a (very) little understanding of what a Shimura variety is, and although I still don't understand the formal definition of that notion, a few vague ideas have come to my mind. Hence ...
9
votes
0answers
85 views

Multiplicity of point as a zero of polynomial.

Let $C \subset \mathbb{P}_2$ be a projective curve defined by a homogeneous polynomial $P(x, y, z)$, and let $L \subset \mathbb{P}_2$ be the line $\{z = 0\}$. Assume $L$ is not a component of $C$. If ...
1
vote
0answers
71 views

A good book to read with Chapter III of Neukirch's “ANT”

The book Algebraic Number Theory from Neukirch is a beautiful book in ANT, but it still have a serious lack in examples and motivation to the concepts. I've already read the first two chapters of the ...
1
vote
2answers
57 views

Manifold over a Finite Field

I'm trying to either associate a manifold with a finite field, or, ideally find a way of considering finite fields as manifolds, in a non-trivial manner. I hope to be able to use this to extend ...
1
vote
1answer
58 views

Vector bundles on $\mathbb{A}^1_k$ with doubled origin?

One of the most common examples of gluing affine lines is the affine line $\mathbb{A}^1_k$ with doubled origin. Out of curiousity, is there a known classfication of the vector bundles on this space?
0
votes
0answers
36 views

Gieseker's theorem for surfaces of general type

I'm trying to understand what is the geometrical meaning of the Gieseker's Th. that is There exist a quasi projective moduli space for surfaces of general type with fixed invariants $K^2$ and $\chi$. ...
2
votes
1answer
45 views

Morphism and Composing morphisms of Varieties

Hi guys I am trying to convince myself that composition of morphisms is again a morphism. If $\phi: V \rightarrow W$ and $\psi : W \rightarrow Z$ are morphisms of varieties. Then $\psi \circ \phi : V ...
0
votes
1answer
24 views

Polynomial approximation on affine varieties

Let $V,W \subseteq \mathbb{A}^n$ be two affine varieties over an algebraically closed field $k$ of characteristic zero and let $a,b\in k$. Q: Can we find a polynomial $f \in k[X_1,...,X_n]$ such ...
1
vote
1answer
54 views

Euler characteristic of a singular fiber

I am trying to understand Kodaira's classification of fibers. In the table at page 41 of Miranda's book http://www.math.colostate.edu/~miranda/BTES-Miranda.pdf there is given the Euler number of the ...