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
votes
2answers
108 views

A particular closed subscheme

Look at the following definition: Let $(X,\mathscr O_X)$ be a scheme. A closed subscheme of $(X, \mathscr O_X)$ is a scheme $(Z, \mathscr O_Z)$ such that: $Z$ is a closed subset of $X$ ...
9
votes
1answer
420 views

Learning Roadmap for Borel - Weil - Bott Theorem

Next semester I may study a course where the ultimate goal is to get to the Borel - Weil - Bott (BWB) Theorem, if not at least try to understand it in the case that we have $G = \text{SL}_n$. I have ...
7
votes
1answer
325 views

Toric Varieties: gluing of affine varieties (blow-up example)

Let $\Delta$ be a fan, consisting of cones $\sigma_0=conv(e_1,e_1+e_2)$ and $\sigma_2=conv(e_1+e_2,e_2)$ and $\tau=\sigma_0\cap\sigma_1=conv(e_1+e_2)$. The dual cones are $\sigma_0^\vee= ...
5
votes
1answer
159 views

Finding the completion of a coordinate ring

Consider $A=\mathbb C[x,y]/(y^2-x(x+1))$, and consider the $\mathfrak m$-adic completion, where $\mathfrak m =(x,y)$. I want to show that this completion is isomorphic to $\mathbb C[[u,v]]/(uv)$, ...
3
votes
1answer
229 views

Why 7 points on a twisted cubic is enough to fix a quadratic?

From Joe Harris, Algebraic Geometry, Page 10. Show that if seven points $p_{1},\cdots,p_{7}$ on a twisted cubic curve, then the common zero locus of the quadratic polynomials vanishing at the ...
1
vote
2answers
314 views

Definition of Zariski Topology

Could someone explain to me, what is Zariski Topology? Under what condition a topology can be called Zariski Topology? Between the set $$V(E)=\{P \in \mathrm{Spec}(R)|E \subseteq P\}$$ and $$D(r)=\{P ...
10
votes
1answer
189 views

Why does Mumford want to avoid “reduction to Jacobians”?

In the introduction to his Abelian Varieties book, David Mumford writes: I don't believe the word "Jacobian" is ever used in this book. Rather stubbornly I wanted to prove that the theory of ...
6
votes
0answers
361 views

sheaf of differential forms - tangent sheaf [Hartshorne]

I'm reading section 8 Differentials of chapter 2 in Hartshorne. It's is extremely hard to me to understand the nature of the definitions: module of relative differential forms - sheaf of relative ...
0
votes
1answer
160 views

Points from an affine subspace with equal distance from given points

Given vector space $\mathbb{R}^3$ with dot product defined as $x \cdot y = 2x_1y_1 + 3x_2y_2 + x_3y_3$ where $x = (x_1,x_2,x_3),y = (y_1,y_2,y_3)$ and given an affine subspace $W: x - y - z - 2 = 0$ . ...
4
votes
1answer
141 views

Intersections of tangents with cubic are colinear

I am trying to do Exercise 5.33 on page 64 of Fulton's book on algebraic curves. 5.33 Let $C$ be an irreducible cubic, $L$ a line such that $L\bullet C = P_1+ P_2 + P_3,$ $P_i$ distinct. Let $L_i$ ...
5
votes
1answer
167 views

The regular representation for affine group schemes

I want to understand the regular representation of an affine algebraic group. An affine algebraic group as I know it, is a functor from the category of $k $ -algebras to groups that is representable ...
3
votes
1answer
786 views

Projective Normality

What is the significance of studying projective normality of a variety ? How does it relate to non-singularity, rationality of a variety ?
26
votes
2answers
586 views

Is there an exposition of complex analysis firmly separating the algebra, analysis, and topology?

Complex analysis seems to work because of the interplay between algebraic geometry over $\mathbb{C}$, and analysis and topology exploiting the fact that $\mathbb{C}/\mathbb{R}$ happens to be a ...
6
votes
1answer
277 views

The distinguished open sets are affine subschemes

If $Y=$ Spec$A$ is an affine scheme and $D(f)\subseteq Y$ (with $f\in A$) is a distinguished open, I want to show that $(D(f),\mathscr O_{Y|D(f)})$ is an affine scheme. Below there is my attempt of ...
11
votes
2answers
815 views

Is there a more elementary proof of this special case of Riemann-Roch?

I'm looking for an elementary proof of the fact that $\ell(nP) = \dim L(nP) = n$, where $L(nP)$ is the linear (Riemann-Roch) space of certain rational functions associated to the divisor $nP$, where ...
1
vote
1answer
62 views

proving that, if finite, the real variety is equal to the complex variety of a real radical ideal

Let $I$ be an ideal of $\mathbb{R}[x_1,\cdots,x_n]$. The real radical $^{\mathbb{R}}\sqrt{I}$ of $I$ is defined as the set of all polynomials of the form $p^{2m}+\sum_{j \in J} q_j^2$, with $p \in I$, ...
4
votes
2answers
406 views

Injective map on Coordinate rings implies dense image?

Let $V$ and $W$ be (irreducible algebraic) varieties over an algebraically closed field $k$. Recall $X\subset W$ is dense if and only if $V(I(X))=W$. Let $f:V\rightarrow W$ be a morphism. If the ...
3
votes
1answer
199 views

Morphism of finite type between affine schemes is quasi-projective

I want to prove that given $A \to B$ a ring homomorphism of finite type, then the induced morphism of schemes $X \to Y$ is quasi-projective. A morphism is quasi-projective if it factors into an open ...
-1
votes
1answer
249 views

Polynomials are continuous with respect to the Zariski topology

Let $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}=\mathbb{C}$. Show that if both $\mathbb{F}$ and $\mathbb{F}^n$ have the Zariski topology then all polynomials on $\mathbb{F}^n$ are continuous.
18
votes
2answers
358 views

What is the difference between $\ell$-adic cohomology and cohomology with coefficient in $Z_\ell$?

Let $X$ be a non-singular projective variety over $\mathbb{Q}$. Consider on the one hand $H^i_B(X(\mathbb{C}),\mathbb{Z}_\ell)$ the singular cohomology with value in $\mathbb{Z}_\ell$, and on the ...
3
votes
1answer
77 views

Smallest projective subspace containing a degree $d$ curve

Is it true that the smallest projective subspace containing a degree $d$ curve inside $\mathbb{P}^n$ has dimension at most $d$? If not, is there any bound on the dimension? Generalization to ...
7
votes
2answers
460 views

Divisors in an abelian surface

How to compute the Néron-Severi group of the abelian surface $Y = \mathbb{C}/\mathbb{Z}[i] \times \mathbb{C}/\mathbb{Z}[i]$. More generally, are there any result that compute the Néron-Severi group of ...
7
votes
1answer
130 views

Helly's Theorem for Biconvex Sets

Helly's Theorem states the following. Suppose that $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq ...
0
votes
1answer
163 views

Find a regular homotopy

Firstly, we define a regular homotopy between regular closed curves as a continuous map $F:$I x I$\rightarrow \mathbb{R}^n$ satisfying the following conditions: (i) for each fixed u $\in$ I, the map ...
2
votes
0answers
123 views

Extending global sections uniquely when extending a line bundle

Suppose we have a family of smooth curves degenerating to a singular one. We might be interested in extending a $g_d^r$ on the (smooth) generic fiber of the family over the singular fiber. So let us ...
4
votes
1answer
170 views

Morphisms between locally ringed spaces and affine schemes

I need some hints to understand the conclusion of the proof of the following lemma from the Stacks Project: Lemma $\mathbf{6.1.}\,$ Let $X$ be a locally ringed space. Let $Y$ be an affine scheme. ...
1
vote
0answers
45 views

Intersection of a variety with an open

Let $V$ be a variety and $U$ an open of $V$. Is the following statement true? If $V \cap U$ is irreducible, then $V$ is irreducible.
11
votes
2answers
548 views

Variety of Nilpotent Matrices

Let $k$ be an algebraically closed field and view $M_n(k)$ as $\mathbb{A}^{n^2}$. $A\in M_n(k)$ is nilpotent if and only if $A^n=0$. Since the equation $A^n=0$ is given by $n^2$ polynomial ...
1
vote
0answers
56 views

Hypersurface containing a component of a variety of higher degree.

Let $X \subset \mathbb{CP}^N$ a $k$-dimensional projective variety cut out by polynomials of degree $\leq d$ and $f_0,\cdots,f_N$ be homogeneous polynomials of degree $d$ without commom factors with ...
1
vote
0answers
101 views

trisecant lemma (reference + result over the real field )

I found the following result in an unpublished lectures notes Let $\mathbb X\subset P^N$ be a non degenerate subvariety of codimension $l>k$. Then the general $(k+1)$-secant $\mathbb P^k$ is not ...
3
votes
1answer
271 views

Is the join of two irreducible varieties is irreducible? (reference + real fied)

The following definition and theorem are taken from J.M. Landsberg, Tensors: Geometry and Applications, Graduate Studies in Mathematics, v. 128 (p. 118) Definition 5.1.1.1: The $join$ of two ...
5
votes
1answer
82 views

An automorphism acting on the sheaf of differentials

I am trying to do the following problem, taken from Iitaka's "Algebraic Geometry". Let C be a smooth, geometrically connected curve of genus g over a field k. Assume that $g \geq 2$. If $f \in ...
4
votes
1answer
84 views

Geometric similarities between points in an algebraic variety

If $f : \mathbb{R} \to \mathbb{R}$ is a univariate irreducible polynomial, Galois theory says that all roots are equivalent up to field automorphism (specifically, an automorphism of the field ...
2
votes
1answer
65 views

What does it say about a multivariate polynomial to be zero on a linear subspace?

If I univariate polynomial $f(x)$ that vanishes at a point $x_0$, we conclude that $x - x_0$ divides $f(x)$, and in particular that $f$ is reducible if $\deg f > 1$. Can anything of significance ...
9
votes
1answer
487 views

Maximal ideals in polynomial rings over a field

Let $K$ be an algebraically closed field and let $k$ be a subfield of $K$ such that the field extension $K \mid k$ is algebraic. Let $B$ be the polynomial ring $K [x_1, \ldots, x_n]$ and let $A$ be ...
1
vote
0answers
205 views

Motivation behind definition of normal domain

An integral domain $A$ is called normal if it equals its integral closure in its field of fractions K. I can appreciate the condition of a ring being equal to its integral closure inside some ring, ...
5
votes
1answer
225 views

Reduction modulo $p$ in number fields

For every prime number $p$, there exist a map $$f:\mathbb{P}^n(\mathbb{Q})\to\mathbb{P}^n(\mathbb{F}_p)$$ defined by: for $P\in \mathbb{P}^n(\mathbb{Q})$, we can find a unique tuple ...
5
votes
3answers
138 views

Are there non-continuable functions that become continuable when raised to some power?

Let $X$ be a complex algebraic variety (integral, separated scheme of finite type over $\mathbb C$) and $U\hookrightarrow X$ an open subvariety. I will say that $f\in\mathscr O_X(U)$ is continuable if ...
3
votes
0answers
96 views

Sheaf cohomology in non-commutative setup

Let $X$ be a topological space and $A$ a sheaf of noncommutative associative algebras over a fixed field $k$. My questions are: 1) Does the category of modules over A have enough injective? 2) If we ...
3
votes
1answer
423 views

When a quotient of a UFD is also a UFD?

Let $R$ be a UFD and let $a\in R$ be nonzero element. Under what conditions will $R/aR$ be a UFD? A more specific question: Suppose $R$ is a regular local ring and let $I$ be a height two ...
4
votes
2answers
168 views

Kernel of a morphism between coherent sheaves.

Throughout this book, http://books.google.co.kr/books/about/Algebraic_Geometry_and_Arithmetic_Curves.html?id=uaLKdA0PxS4C&redir_esc=y, kernel of a morphism between coherent sheaves on a locally ...
0
votes
1answer
117 views

Prove that the tetragon ABCD also has an inscribed circle

There is given convex tetragon ABCD and some points J,K,L,M on the line segments AB,BC,CD,DA respectively. The intersection of JL and KM is T. Each of the tetragons AJTM,BKTJ,CLTK, and DMTL have an ...
4
votes
1answer
201 views

Classifying map

Let $\xi=(E,p,B)$ a principal $G$-bundle and $\eta=(P,\pi,Q)$ a real vector bundle such that $\operatorname{rank}(\eta)=n$. We can consider a classificant space $BG$. What is the classifying map $f:X ...
5
votes
2answers
256 views

Self-intersection number of a complex curve in complex projective space

I'm currently trying to get a grip on actually calculating some differential-geometric definitions. I'm looking at the following map from $\mathbb{CP}^{1}$ to $\mathbb{CP}^2$ : ...
4
votes
1answer
58 views

Quasicompact over affine scheme

Let $X$ be a scheme and $f : X \rightarrow \mathrm{Spec}\, A$ a quasicompact morphism. Are there any easy conditions on $A$ under which we can say that $X$ is quasicompact? Quasicompact morphism ...
4
votes
1answer
145 views

How many types of surface singularities multiplicity two exist?

All varieties are over $\mathbb{C}$. Let $S$ be a reduced algebraic surface in $\mathbb{P}^3$ with a singular point $p$ of multiplicity two. The question is local so we reduce to $S \subset ...
3
votes
1answer
70 views

Where to find the proof for the isomorphism between the (2n-1)_th homotopy group of Aut(V) to Z?

I am learning Bott-Seeley's comment article on the Callias' index theorem. They quoted a theorem, that is, $\pi_{2n-1}(Aut(V_{C}^{n}))\simeq Z$ or $\pi_{2n-1}(GL_n(C))\simeq Z$. Can you tell me the ...
3
votes
1answer
98 views

Is there an open mapping theorem for affine morphisms?

Let $A$ and $B$ be rings. If $\varphi : A \longrightarrow B$ is such that $^a\varphi : Spec(B) \longrightarrow Spec(A)$ is bijective, then in what conditions $^a\varphi$ is a homeomorphism? Or, more ...
7
votes
0answers
159 views

Why do algebraic varieties contain curves passing through two given points

Let $X$ be an algebraic variety over the complex numbers. My definition of an algebraic variety is a finite type separated $\mathbf C$-scheme. Someone told me that such varieties have the following ...
3
votes
0answers
59 views

Maps between total spaces of holomorphic vector bundles

I am wondering what is possible and what is not possible regarding maps between the total spaces of holomorphic vector bundles. Let me outline a situation that is a bit more concrete, to help focus ...