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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1answer
35 views

The map $f\colon\mathbb{A}^2_k\to\mathbb{A}^2_k$ given by $f(x,y)=(x,xy)$ is birational?

I'm reading a bit about rational maps, and I'm still trying to get get my head around birational maps. Consider the map $f\colon\mathbb{A}^2_k\to\mathbb{A}^2_k$ on the affine $2$-space over $k$ ...
4
votes
1answer
35 views

Finding Basis for a Radical of an Ideal

I am to find a basis of the following ideal: $$\sqrt{<x^5-2x^4+2x^2-x, \quad x^5-x^4-2x^3+2x^2+x-1>}$$ Truth be told, I'm not entirely confident of my solution. I will present it and then ask ...
6
votes
0answers
132 views

Andre-Quillen Homology of the cuspidal curve $k[x,y]/(x^2 - y^3)$

I was wondering if I am in the right track here. Let $A := k[x,y]/(x^2 - y^3)$, the cuspidal curve. Obviously this isn't etale or smooth over $k$ so its cotangent complex is not contractible. Now, I ...
2
votes
1answer
64 views

9 missing lines on a specific smooth cubic surface

Let $\Gamma (x,y,z) = 27 x^3 + 243 x^2 y+324 x y^2 + 189 y^3 +27 x^2 z + 27 x y z - 27 y^2 z + z^3$. $S: \Gamma (x,y,z) = 27 $ is a smooth cubic surface. Consider lines of the form $x = x_0 + p s$, $y ...
0
votes
1answer
43 views

A birational map from $\mathbb{P}^1$ to an irreducible plane projective curve

Let $C$ be an irreducible plane projective curve described by the equation $$zf(x, y) + g(x, y) = 0,$$ where $f$ and $g$ are a homogenous forms of degree $d - 1$ and $d$, respectively. What would be ...
2
votes
1answer
67 views

Galois theory on curves

Context: Let $\mathbb{F}$ be the algebraic closure of $\mathbb{F}_q$ for $q$ prime. We know that $\mathbb{F}(t)$ for $t$ transcendental is the function field of the projective line ...
1
vote
1answer
46 views

The affine line with two points removed

To which affine variety $V$ is $\mathbb{A}^1 \setminus \{0, 1\}$ isomorphic to? What would be the isomorphism in this case? Any help would be appreciated.
3
votes
0answers
40 views

Applications of resolution of singularities

I would to know applications of Resolution of Singularities, this means what is profits of having a resolution of singularities of a variety both in and out of mathematics and both in positive and ...
0
votes
0answers
34 views

Zariski's lemma

I have a question concerning Zariski's lemma: is it still true if we assume the ground field to be a finite one? $k \subseteq K$ field extension, with $k$ a finite field (of $p^n$ elements) and $K$ ...
1
vote
1answer
35 views

Question on Generic Freeness, ref. [Matsumura, page 185]

I am sure this must have been answered somewhere but I can't find them, so I shall try my luck here. Let $A$ be a Noetherian integral domain and $M$ a finitely generated $A$-module. Then there ...
1
vote
1answer
25 views

Self-Studying Algebraic Geometry: Finding $f$ in $I(V(J))$

Problem 4.1.2 in Ideals, Varieties, and Algorithms asks: Let $J=\langle x^2+y^2-1,y-1\rangle$. Find $f \in I(V(J))$ such that $f \notin J.$ I started by trying to get an idea of what $V(J)$ is- ...
0
votes
2answers
63 views

What is $S_d$ in algebraic geometry?

I'm trying to read algebraic geometry on my own by doing homeworks on course hold in 2003. One of the problem is the following: Let $k$ be a field, $S=k[T_0,\ldots,T_r]$, ...
1
vote
1answer
34 views

Construction of Spec$(\mathcal{B})$ on an affine base scheme.

I apologize for the title, but it was difficult to find a more appropriate one, feel free to edit if you think it is necessary. My questions comes as part of the proof that, for a scheme $X$, and a ...
1
vote
1answer
43 views

Computing Resultant of Two Polynomials

Consider these two polynomials: $$f=x^2y+3xy-1$$ $$g=6x^2+y^2-4$$ I need to compute their resultant, denoted in my textbook as $h=Res(f,g,x)$. Here's where I need help: setting up the Sylvester ...
3
votes
0answers
36 views

Henselization of the ring of polynomials

I am trying to understand example of Henselization from wiki. http://en.wikipedia.org/wiki/Henselian_ring#Henselization It says that Henselization of the ring of polynomials localized at point $(0, ...
3
votes
0answers
36 views

Unwinding descent via Barr-Beck

Let $f: U \rightarrow X$ be a faithfully flat morphism of nice schemes (quasiseparated, quasicompact, and anything else I might have forgotten). One can understand descent in quasicoherent sheaves ...
0
votes
1answer
40 views

Generic flatness on modules

I am looking for a stronger notion of generic flatness. Let $A$ be a Noetherian ring, $M$ a finitely generated module over $A$. Suppose there exists a maximal ideal $m$ of $A$ such that $M_m$ (the ...
0
votes
0answers
35 views

Leray-Hirsch analogue of algebraic geometry

I want to use a Leray-Hirsch analogue of algebraic geometry to construct the chern classes. I am not sure how to prove the statement. Suppose that $E$ is a locally free sheaf of rank $r$ on $X$. I ...
1
vote
0answers
59 views

Maximum product of lengths involving secant drawn to a parabola.

A chord is drawn from a point $P(1,t)$ to the parabola $y^2=4x$, which cuts the parabola at $A$ and $B$. If $PA\cdot PB=3|t|$, what is the maximum possible value of $|t|$? All I can infer is that the ...
1
vote
1answer
33 views

Representable open immersion of functors is a monomorphism

I have a question concering the proof of theorem 8.9 in Algebraic Geometry I (U. Görtz, T. Wedhorn). I will introduce what is needed. Let $S$ be a scheme, for an $S$-scheme $X$, we denote ...
3
votes
3answers
167 views

Problem in proving that $\mathbb{A}^2$ is not homeomorphic to $\mathbb{P}^2$

let $k$ be an algebraic closed field. All the spaces are equipped with the usual zariski topologies. All the proofs of this fact that I've seen rely on the fact that two lines in $\mathbb{P}^2$ ...
2
votes
0answers
28 views

could someone help me with this particular detail in this article?

I'm reading this article and I was stuck in this part: I didn't understand his trick computing the orders at $P$. Remark: He defines $D=\inf\{div(\omega_{g-1}),div(\omega_g)\}$, i.e., ...
3
votes
1answer
77 views

Understanding the stack $B\mathbb{Z}$

Here, let $\mathbb{Z}$ be the group scheme whose functor of points is the constant functor which takes a connected affine scheme to the group $\mathbb{Z}$. I'm having a bit of trouble understanding ...
2
votes
1answer
35 views

Identifying the orbit space of the unitary group $U(n)$ in the compact symplectic group $Sp(n)$

Let $Sp(n)$ be the compact symplectic group. Let $U(n)$ the unitary group, and $O(n)$ the orthogonal group. What is $Sp(n)/U(n)$? What is $U(n)/O(n)$? I obtain that ...
5
votes
1answer
49 views

Fixed point of curve automorphism

Exercise I.F-8 from "Arbarello, Cornabla, Griffiths, Harris: Geometry of algebraic curves" states that for a complex algebraic genus $g$ curve and its automorphism $\varphi$ of order $n$ the number of ...
0
votes
0answers
26 views

How to check whether a linear map on integral domains is a formal derivative

I have an elementary question on formal derivatives. Assume $A=K[X,Y,Z]/I$ is an integral domain (for example $I$ is a prime ideal and K is the field of rationals). Let $d:A\to A$ be a linear map. Is ...
1
vote
1answer
27 views

connection between discrete valuation rings and points of a curve.

Let $C$ be a projective irreducible non-singular curve over a field $k$ and let K be its function field. It applies that $(k[X,Y]/I(C))_{(X-a,Y-b)}$ (i.e. the localization of $k[X,Y]/I(C)$ at ...
2
votes
2answers
39 views

If $f\in k(\mathbb{A}^1)$ and $f^2\in k[\mathbb{A}^1]$, then is $f\in k[\mathbb{A}^1]$?

Suppose $k$ is algebraically closed, and $f\in k(\mathbb{A}^1)$ is in the field of rational functions over the variety $\mathbb{A}^1$. If we also know that $f^2$ is in the coordinate ring ...
2
votes
1answer
41 views

Restrictions of maps between projective varieties.

Let $f\colon X\to Y$ be a surjective algebraic map between two projective $k$-varieties, where $k$ is algebraically closed. Let $n=\dim(X),\,m=\dim(Y)$. Suppose furthermore that X,Y are irreducible. ...
8
votes
2answers
231 views

Is the ring of p-adic integers of finite type over the ring of integers?

Denote by $\mathbb{Z}_p$ the ring of $p$-adic integers. Is $\mathrm{Spec}(\mathbb{Z}_p)$ of finite type over $\mathrm{Spec}(\mathbb{Z})$?
2
votes
0answers
22 views

another representation of the zeta function of a curve over a finite field

Let $C$ be a non-singular curve over $\mathbb{F}_q$. Denote by $d$ the degree map from the group of divisors to $\mathbb{Z}$ and denote by $P$ the set of prime divisors w.r.t. to the function field. ...
1
vote
2answers
49 views

examples of interpreting schemes (Eisenbud)

I am having trouble understanding the role primary decomposition plays in ``interpreting'' the geometric picture of a scheme. Here are the examples I am struggling with from Eisenbud's Commutative ...
4
votes
1answer
26 views

coefficients of the zeta function of curve over a finite field $\mathbb{F}_q$

Let $C$ be a non-singular curve over $\mathbb{F}_q$. Denote by $d$ the degree map from the group of divisors to $\mathbb{Z}$ and denote by $P$ the set of prime divisors w.r.t. to the function field. ...
0
votes
1answer
18 views

Is the closed subgroup of any semisimple linear algebraic group semisimple?

Let $S$ be a semisimple linear algebraic group $/K$, with $K$ a field and $char K = 0$. Let $H \leq S$ be a closed subgroup $/K$. Is $H$ semisimple?
1
vote
0answers
34 views

Chow lemma + resolution of singularities

I am trying to understand the following simple reduction in Deligne's Theorie de Hodge II. He works in the category of schemes of finite type over $\mathrm{Spec}(\mathbf{C})$. Let $f : X \to S$ be a ...
0
votes
0answers
29 views

What is the kernel of $I/I^2 \to \Omega_{\mathbb P^{n}/k} \otimes \mathcal O_X$?

Recall that if $X \subset \mathbb P^n$ is a smooth projective variety, we have the conormal sequence of locally free sheaves on $X$ (here $I$ is the ideal sheaf of $X$): $$ I/I^2 \xrightarrow{\delta} ...
1
vote
1answer
19 views

Subgroup of an affine, algebraic irrducible group.

Let $G\subseteq GL_n(\mathbb{C})$ be a irreducible, affine, algebraic group (Zariski-Closed). Moreover let $H \subseteq G$ be a finite normal group. I want to show that $H \subseteq Z(G):=\{g \in ...
2
votes
1answer
17 views

connected linear algebraic group over the algebraic closure of a field

Let $G$ be a connected linear algebraic group over a field $k$ of characteristic 0. A paper I'm reading seems to imply that $\overline{G}:= G \times_k \overline{k}$ will also be connected, but I don't ...
1
vote
0answers
29 views

A question about Hilbert's Nullstellensatz

We know that Hilbert's Nullstellensatz is valid for $\Bbb{C}[X]$, as $\Bbb{C}$ is a closed field. Let us consider the ideal $(x+y,x-y)\subset \Bbb{C}[x,y]$. Clearly, $Z((x+y,x-y))=\{(0,0)\}$. Now ...
3
votes
1answer
81 views

What is GAGA for dimension 1 ? (Historical Question)

I know Riemann surfaces are actually algebraic curves, i.e. all Riemann surfaces can be simply embedded into some projective space $\mathbb{P}^n$. But this doesn't indicate me more correspondences ...
0
votes
1answer
23 views

Condition for dim of the Euclidean space with orthogonal basis

I would like to show that if the orthogonal basis of the $\Bbb R^n$ Euclidean space with the standard dot product has the vectors whose elements are exclusively $1$ or $-1$, then $n \le 2$ or $n$ is ...
1
vote
0answers
42 views

Projections of a rational normal curve of $\mathbb{P}^4$ (Exercise 3.9 in Harris' _Algebraic Geometry_)

In exercise 3.9 of his Algebraic Geometry book, Prof. Harris asks to show that the rational quartic curves $$C_{a,b}=[X^4-aX^3Y,X^3Y-aX^2Y^2,bX^2Y^2-XY^3,bXY^3-Y^4]$$ are projections of a rational ...
0
votes
0answers
30 views

quadric in $P^{3}$

I have a difficulties to show the followings: Let k be an algebraically closed field. A quadric in $P^{3}(k)$ is a projective algebraic set of the form Q = V(F), where F is an irreducible polynomial ...
2
votes
1answer
43 views

Is the Projectivization of a coherent sheaf on a reduced Noetherian scheme reduced again?

Let $X$ be a reduced Noetherian scheme and $\mathcal{F}$ a coherent sheaf on $X$. What i am wondering is: Is it true or not that the projectivization ...
0
votes
0answers
28 views

Some basic questions about fibered surfaces

I get stuck at the section 8.3 Fibered Surfaces of Qing Liu's book Liu: Algebraic Geometry and Arithmetic Curves and I feel strange that it is not easy to find many other books or papers discussing ...
7
votes
3answers
167 views

What does the Tate module of an elliptic curve tell us?

I started studying elliptic curves, and I see that it is rather common to take the Tate module of an elliptic curve (or, of the Jacobian of a higher genus curve). I'm having a hard time isolating the ...
6
votes
0answers
78 views

Duals of representations of affine group schemes, in particular $\mathrm{GL}_n$

Duals of representations of affine group schemes Let $R$ be a commutative ring. If $G$ is a group and $V$ is a dualizable i.e. finitely generated projective $R$-module on which $G$ acts, then it is ...
2
votes
1answer
45 views

Smallest Convex Hulls ($n$-Simplex) of $n+2$ Points in $\mathbb{R}^n$

"Given $n$, find the minimal value of $k$ with the following property: Any $k$ (distinct) points in $\mathbb{R}^n$ can be partitioned into two disjoint subsets so that the intersection of the convex ...
1
vote
1answer
48 views

Integral extensions with finitely generated k-algebras

I have $k$ a field, and I am assuming that the finitely generated $k$-algebra $K = k[x_1,x_2]$ is also a field. I am trying to prove Zariski's lemma in this case, by seeing first that $K$ is an ...
6
votes
2answers
64 views

Is the set of complex solutions to $x^2+y^2 = 1$ isomorphic to $\mathbb{C}^*$?

In his article about Grothendieck, Edward Frenkel states that the set of complex solutions to the equation $x^2+y^2 = 1$ is "a plane with one point removed." I'm curious how this can be made precise. ...