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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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 ...
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1answer
14 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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26 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 $L, L' \subset \mathbb{P}^3$. ...
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0answers
28 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 ...
2
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0answers
17 views

Intersection points of two Bernoulli lemniscates

What is the maximum number of intersection points of two Bernoulli lemniscates in the real plane? A Bernoulli lemniscate is a degree four curve with two nodes on the line of infinity in complex ...
2
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1answer
30 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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31 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 ...
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36 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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36 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 ...
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2answers
100 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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36 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
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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
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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 ...
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2answers
73 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?
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1answer
21 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. ...
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1answer
50 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 ( ( ...
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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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17 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 ...
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1answer
22 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?
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1answer
41 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 ...
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1answer
14 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 ...
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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 ...
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23 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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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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21 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 ...
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54 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
39 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
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1answer
34 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 ...
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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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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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28 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 ...
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1answer
21 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 ...
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2answers
45 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 ...
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23 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
18 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
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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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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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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}} + ... + ...
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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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31 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
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1answer
46 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 ...
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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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37 views

Find the irreducible components of algebraic curve. [on hold]

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. ...
3
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1answer
48 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 ...
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21 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 $GL$?
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35 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 ...
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74 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 ...
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0answers
50 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 ...