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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7 views

Projective line intersecting 3 projective subspaces

I am trying to solve the following problem: Let $\mathbb{P}(U),$ $\mathbb{P}(V)$ and $\mathbb{P}(W)$ be projective subspaces of dimension $k,$ $l$ and $m$ respectively in $\mathbb{P}_K^n$. Suppose ...
2
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0answers
15 views

Singular Conics and Intersection of Line with a Conic

I've been working through Silverman and Tate's book Rational Points on Elliptic Curves. They use conic equations as an introduction to singular/nonsingular curves. I've reproduced the problem with my ...
1
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1answer
23 views

WLOG doubt: why can we assume that two disjoint linear subspaces of dimension $n$ in $\mathbb{P}^{2n+1}$ are given by the following equations…

Let $H_1,H_2$ be two linear disjoint subspaces of dimension $n$ in $\mathbb{P}^{2n+1}$. Let $(x_0:\cdots:x_n:y_0:\cdots:y_n)$ be homogeneous coordinates in $\mathbb{P}^{2n+1}$. My question is: ...
-1
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1answer
16 views

Crushing Cylindrical wooden pillar

The Crushing load of a cylindrical wooden pillar varies directly as the fourth power of its diameter and inversely as the square of its height. If K is the crushing load of a certain wooden pillar, ...
3
votes
1answer
22 views

Tangent lines of a smooth curve $C \subseteq \mathbb{P}^2$

Let $C$ be a smooth curve, given as the zero locus of a homogeneous polynomial $f \in \mathbb{K}[x_0,x_1,x_n]$. Consider the morphism $\varphi_C:C\rightarrow\mathbb{P}^2$ such that ...
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0answers
17 views

Question about bilinear and quadratics form

I'm reading this book: Geometry of algebraic curves by Cornalba, Harris etc. At page 289 there is an excercise where the authors define a quadratic form $Q:V \times V \rightarrow \mathbb{C}$ taking ...
0
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1answer
16 views

Subgroups of points of order 2 in an elliptic curve

Depending on the roots of $y^2 - x(x^2+ax+b) = 0$ being real or not, we can have 2 subgroups of points of order 2 for a given elliptic curve- Kelin-4 group or a cyclic group of order 2. How does one ...
2
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0answers
42 views

A computation related to Hironaka's Example

My questions are at the very end, first I'll describe the context. Let $f:\mathbb{P}^3\to \mathbb{P}^3$ be an involution whose fixed locus consists of two disjoint lines $L, L' \subset \mathbb{P}^3$. ...
2
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0answers
36 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 ...
3
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0answers
28 views

Intersection points of two Bernoulli lemniscates

What is the maximum number of intersection points of two Bernoulli lemniscates in the real plane? Here is some of my efforts: A Bernoulli lemniscate is a degree four curve with two nodes on the ...
2
votes
1answer
31 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 ...
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0answers
33 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
38 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 ...
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0answers
37 views

Algebraic closedness in residue field [on hold]

If $A\subseteq B$ are affine domains over an algebraically closed field of $k$ of characteristic zero such that $Q(A)$ is algebraically closed in $Q(B)$, how can one show that $Q(A)$ is also ...
2
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2answers
111 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 ? ...
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0answers
41 views

Tangent Cone of a Complete Intersection

Can you give me an example of an affine variety $X \subseteq \mathbb{A}^n_{\mathbb{C}}$ over the complex numbers which is a complete intersection such that the reduced tangent cone at some point $p ...
5
votes
1answer
35 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
30 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
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2answers
75 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
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1answer
22 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. ...
0
votes
1answer
52 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
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1answer
33 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 ...
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0answers
18 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
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1answer
23 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
42 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
15 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
39 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
24 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 ...
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0answers
13 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
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0answers
22 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
55 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 ...
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1answer
40 views

Intersection of a hypersurface with a projective variety [on hold]

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
35 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
26 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 ...
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0answers
14 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 ...
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0answers
29 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
22 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 ...
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2answers
40 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
51 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 ...
3
votes
0answers
24 views
+50

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 ...
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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 ...
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1answer
19 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
55 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 ...
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0answers
91 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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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
52 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 ...
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0answers
32 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
48 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
13 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 ...
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0answers
37 views

Find the irreducible components of algebraic curve. [closed]

Ley $Y$ be the algebraic set in $\mathbb{A}^3$ defined by the two polynomials $x^2-yz$ and $xz-x$. show that $Y$ is a union of three irreducible components. Describe them and find their prime ideals. ...