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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14 views

Radicial Morphism over DVR's

I would like a reference for the truth/falsity of the following statement: Suppose that $X \rightarrow Y$ is a map of $S$ schemes where $S$ is the spectrum of a DVR with generic point $\eta$ and ...
1
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0answers
19 views

Connectedness of automorphism group of a variety

Let $Y$ be a proper, smooth, integral variety over an algebraically closed field $k$ of characteristic zero. Consider the automorphism group $Aut_k(Y)$ (a group scheme). Are there any natural ...
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27 views

When does a principal divisor have degree 0?

We know that on a smooth curve over an algebraically closed field principal divisors have degree 0. This holds true also for the projective space over any field (in this case equality holds too). My ...
1
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0answers
21 views

Codimension of Schemes

Let $X$ be an integral scheme over a field $k$ of finite type. For a closed non-empty subset $Y$ show that $\operatorname{codim}(Y,X) = \inf \{ \dim \mathcal{O}_y \mid y\in Y \}$. It is easy to prove ...
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20 views

Constructing relatively ample line bundles

Let us work over $\mathbb C$. Let $X$ be a smooth projective variety with an ample line bundle $L$. Let $S$ be any scheme over $\mathbb C$, reduced or integral (or... add more assumptions if ...
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16 views

Formal smoothness of $A \to A[t]/(h)$.

Let $A$ be a commutative noetherian ring, $T$ an indeterminate, $h=h(T) \in A[T]$, and $B:= A[T]/(h)$. When $B$ is formally smooth over $A$? (If $h$ is monic, is $B$ formally smooth over $A$?). ...
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0answers
17 views

Self-intersection of the exceptional curve of blown up of a cone at vertex

Suppose $V=V(X^2+Y^2=Z^2)$ is the projective completion of affine cone in $\mathbf{P}^3=\textrm{Proj} \ k[X,Y,Z,W]$, the vertex $P=(0,0,0,1)$ is singular, if we blow up $P$ we get a exceptional ...
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0answers
19 views

When are the integral cohomology groups of a toric variety free?

Let $X=X_{\Sigma}$ be a complete, toric variety associated to the fan $\Sigma$. I am interested in conditions on $\Sigma$ (or on the polytope $P$ when $\Sigma$ is the normal fan of $P$) that guarantee ...
5
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0answers
40 views

Explicit examples of (co)limit arguments in other fields

Over the past weeks, I have noticed that high level lecture notes in subjects like algebraic geometry, algebra, and algebraic topology often sketch proofs in the following form: Proof sketch ...
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0answers
15 views

book suggestions on homogeneuos coordinate and projective geometry [on hold]

Will any one introduce some good books about projective geometry and homogeneuos coordinates to me?
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0answers
14 views

Smooth section of Hodge bundle ($F^pH^k$) can be viewed as a smooth form of type$F^pH^k(X,C)$ over$ X$,$ X--->B$ is an analytic family.

I think it is due to Kodaira. could someone explain the idea that Kodaira come up with this. maybe I shouldn't say"can be viewed as". I really mean the smooth form restrict on each fibre is just the ...
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29 views

When the fibers of $A \to A[T]/(h(T))$ are geometrically regular?

Let $A$ be an affine commutative noetherian domain over a characteristic zero field $K$, $T$ an indeterminate, $h=h(T) \in A[T]$ (not necessarily monic), $B=A[T]/(h)$, and assume that $(h)$ is a prime ...
-1
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0answers
30 views

Is Tu's “Introduction To Manifolds” a good place to pick up diff-geo intuition for Vakil's notes?

So I want to study algebraic geometry from Ravi Vakil's notes. However, the only thing I seem to be missing -- I have all the official prerequisites like commutative algebra and point-set topology ...
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0answers
19 views

Using Magma to solve a multivariate polynomial system with parameters

I want to solve a system of multivariate polynomials with parameters. Mathematically, the ground field is F = Q(a, b, c, …), the field of rational functions. The polynomials are in F[x,y,z,…]. I ...
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0answers
25 views

In the triangle ABC, if BC = [2 [(AC)^2-(AB)^2]/[-AC + sqrt[(AC)^2+4 (AB)^2], prove that 3m(<C) = 2m(<B) [on hold]

Given a triangle ABC, if $$a = \dfrac{2(b^2 - c^2)}{-b + \sqrt{b^2 + 4c^2}}$$, prove that $3m(C) = 2m(B)$.
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0answers
39 views

Short exact sequence of groups schemes and dimensions

Let $G$ be a projective groups scheme over an algebraically closed field of positive characteristic $p$. Denote by $G_t$ the $p$-torsion part of $G$ i.e., elements $g \in G$ such that $g^p=0$. Is ...
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3answers
71 views

Describe $Spec( \mathbb C[x,y]/x(x-a))$ where $a$ is some complex number.

I'm solving some exercises from my class notes on Commutative Algebra,On the following exercise I got stuck: Describe $Spec( \mathbb C[x,y]/x(x-a))$ where $a$ is some complex number. As far as I ...
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0answers
20 views

For a subspace $S $ in higher dimensions, what is $|S| $?

Let $S\subset \Bbb {A}(k) $. What is $|S|$? I came across this notation whilst studying Algebraic Geometry (conditions imposed by $S $ on polynomials of degree $\leq d $). Thanks!
2
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0answers
86 views

What would the ring $\mathbb{Z}[x,y]/(x^2-y)$ look like?

What would the ring $\mathbb{Z}[x,y]/(x^2-y)$ look like? And also what would the ring $\mathbb{Z}[x,y]/(x^3-x-y^2)$ look like? These are two sorts of rings I have been curious about.
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24 views

Use of discriminent in proving that the points of unramification is open…

I am confused about Shaferevich Varieties in Projective Space proposition 2.29: If $f : X \to Y$ is a finite map between irreducible varieties, with $Y$ normal, then the set of points in $Y$ over ...
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0answers
26 views

example of computing ramification index

I am trying to understand example 2.2.9 of Silverman's "Arithmetic of Elliptic Curves". In this example, Silverman considers a map $$ \phi:\mathbb{P}^1\to \mathbb{P}^1; [X,Y]\mapsto [X^3(X-Y)^2, ...
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0answers
38 views

Is $\mathbb{G}_{m,k}$ (the multiplicative group) simply connected?

I have a field $k$ (which I can take to be algebraically closed if it makes the answer simpler) with the char $k = 0$. The multiplicative group $\mathbb{G}_{m,k}$ is $spec (k [x, x^{-1}])$. ...
3
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0answers
62 views

Fulton Algebraic Curves, Exercise 2.32(a).

Let $R$ be a DVR satisfying the conditions of Problem 2.30. Any $z \in R$ then determines a power series $\lambda_i X^i$. If $\lambda_0, \lambda_1, \dots$ are determined as in Problem 2.30(b). Show ...
3
votes
3answers
66 views

Trying to Understand a Remark about Zariski Topology

I'm reading some notes in which following remark is given: The Zariski topology is quite different from the usual ones. For example, on affine space $ \mathbb A^n$ a closed subset that is not ...
2
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0answers
24 views

Tangent to dual curve.

Let $C$ be a smooth projective plane curve, let $P \in C(k)$, and let $\ell$ denote the tangent line to $C$ at $P$. Let $C^*$ denote the dual curve to $C$, in the dual plane $(\mathbb{P}^2)^*$ (the ...
3
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2answers
64 views

Show $k[x]\cap(y-x^2,z-x^3)={0}$ in $k[x,y,z]$. $ k$ field.

Let k be a field. How could I show that $k[x]\cap(y-x^2,z-x^3)={0}$ in $k[x,y,z]$. I understand that there's a whole algorithm I could go through with Grobner basis, elimination theorem etc. but ...
0
votes
1answer
45 views

When is the geometric Picard group $Pic(X_{\overline{K}})$ of finite type?

Let $X$ be a smooth proper geometrically connected variety over a field $K$ of characteristic 0. Let $\overline{K}$ denote an algebraic closure of $K$. What other conditions on $X$ are needed so ...
0
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0answers
43 views

Tangent space of an algebraic variety $ X $ in a point $ x \in X $.

Let $X$ be an algebraic variety, and $ \mathcal{O}_{X,x} $ its local ring at a point $x \in X$, and $ \mathfrak{m}_x $ its maximal ideal. Let set $ k_x = \mathcal{O}_{X,x} / \mathfrak{m}_x $ the ...
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0answers
38 views

Is a local equation for a smooth point on a curve given by the equation for the “tangent line”?

Let $X$ be an algebraic curve in $A^m$ defined by some equations $F = (f_1, \ldots, f_n)$. If $p$ on $X$ is a smooth point, general nonsense guarantees that there is a local equation for $p$. Is this ...
0
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0answers
25 views

Degree of vector bundle under pushforward while blowing up

Let $f:X\longrightarrow Y$ be a birational morphism of projective varieties over $\mathbb{C}$. In particular we can assume that $X$ is a blow of $Y$ at finitely many points. Let $F$ be a vector bundle ...
1
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2answers
24 views

Parametrization of a sphere

I am trying to argue geometrically that mapping the point $(u,v,0)$ to $(x,y,z)$ gives a parametrization of the sphere $x^2+y^2+z^2=1$ minus the north pole. My questions are: a) What exactly is a ...
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0answers
35 views

Barring a morphism to subvarieties

This is exercise I.3.10 from Hartshorne.I understand that restrict a morphism is continuous but not understand the topological structure of a locally closed irreducible in connection with regular ...
2
votes
1answer
58 views

Proof of Chow's lemma in EGAII

Section 5.6 of EGAII is dedicated to Chow's lemma. I am having a hard time following an early step of the proof. The version of Chow's lemma in the text assumes that $X$ is a separated $S$-scheme of ...
1
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2answers
29 views

Dickson's Lemma (proof of Prop. 2.23 in Hasset's Intro to Alg Geom)

I'm studying Hasset's book by myself but I had no previous formal algebra training. To prove Dickson's lemma (prop. 2.23, p. 19) he defines the auxiliary monomial ideals $$J_m=\left<x^\alpha \in ...
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0answers
26 views

How to find location - multilateration

I have this data: $$ {x1} = 473463,100288[m]\\ {y1} = 5924242,046998[m]\\ {z1} = 0[m]\\ {t1} = 41919,84025[s]\\ {x2} = 473483,237020[m]\\ {y2} = 5924212,730018[m]\\ {z2} = 0[m]\\ {t2} = ...
2
votes
0answers
46 views

Line bundles on $\mathbb{P}^2 \times \mathbb{P}^2$

I'm curious as to what are all the line bundles on $\mathbb{P}^2 \times \mathbb{P}^2$? I might be mistaken but I believe they are classified as $$ \mathcal{O}_{\mathbb{P}^2 \times \mathbb{P}^2}(p,q) ...
2
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0answers
26 views

Abelian hypersurfaces

Is there any way to tell from its defining equation when an affine hypersurface has a group law? I am aware that all such groups are matrix groups; in that case I might want to ask when a variety over ...
3
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0answers
40 views

The symmetric product of elliptic curves

Suppose $E$ is an elliptic curve, what is the symmetric product $F=E\times E/S^2$? It is a smooth surface, let $\pi\colon E\times E\to F$ be the projection, then we have ...
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0answers
36 views

Automorphisms of cubic nodal curve

How to calculate the automorphism group of the nodal cubic curve $y^2=x^3+x^2$ ? Should I use the rationality of this cubic curve ?
2
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0answers
69 views

When is “number of points in the fiber” semicontinuous?

Let $f:X \rightarrow Y$ be a finite morphism of schemes. For $y \in Y$ let $n(y)$ be the cardinality of the set-theoretic fiber $f^-1(y)$. This is finite since $f$ is finite. I think that the ...
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0answers
31 views

Global section on product of sheafifications

Say I have two presheaves $\mathcal F$ and $\mathcal G$ of $\mathcal O_X$-modules on a scheme $X$. Let $\mathcal F^+\!, \mathcal G^+\!$ be their respective sheafifications. We then have a commitative ...
2
votes
1answer
44 views

Is the quotient morphism from product of curves to to their symmetric product flat?

Suppose $C$ is a smooth curve, is the morphism $C^2=C\times C\to C^{(2)}=C\times C/S_2$ flat? What about the general case?
2
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0answers
36 views

Blowing up fibers in families - looking for comparison results

Given a morphism $\pi: S\rightarrow B $, an ideal sheaf $I$ on $S$, and a point $b\in B$, I wish to consider the blow up of $S$ along $I$. Say I know something about the pullback $I(b)$ to the fiber ...
0
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0answers
39 views

Simple question about closed immersions

I would like to know if I understand closed immersions correctly. Let $f:X \rightarrow Y$ be a morphism of schemes (or locally ringed spaces in general). The morphism $f^\sharp:\mathscr{O}_Y ...
1
vote
1answer
64 views

Dimension of $\mathfrak{m}^k/\mathfrak{m}^{k+1}$?

Let $C \subset \mathbb{P}_\mathbb{C}^2$ be a projective curve, and $p \in C$ a point of multiplicity $m$. If $\mathcal{O}_p(C)$ is the local ring of $C$ at $p$, and $\mathfrak{m} \subset ...
4
votes
1answer
97 views

Galois invariants of the Tate module of an elliptic curve over a number field

Let $K$ be a number field, $E$ be an elliptic curve over $K$, $l \neq p$ be two different prime numbers and $v$ be a place of $K$ above $l$. I am trying to understand the proof of proposition I.6.7 ...
3
votes
1answer
30 views

Nonsingular cubic curve, quotient of $d(x/z)$ and $y/z$ is differential which is regular everywhere.

Let $C \subset \mathbb{P}_2$ be the cubic curve defined by$$y^2z = x(x-z)(x-\lambda z)$$with $\lambda \in \mathbb{C} - \{0,1\}$. Let $p = [0, 0, 1]$, $q = [1, 0, 1]$, $r = [\lambda, 0, 1]$, and $s = ...
3
votes
0answers
28 views

Normal bundle of zero scheme of section

Suppose $Y$ is a smooth variety, $E$ is a rank $d$ bundle on $Y$, $s$ is a regular section of $E$ over $Y$,(i.e.,locally under a trivialization $E|_U\cong O_U^d$, write $s=(s_1,\dots,s_d)$, then $s_i$ ...
2
votes
0answers
48 views

Ideals in a ring as geometric objects?

I am interested in learing about the possibility of (one-sided) ideals in a ring being repreented geometrically. In other words, about their status as geometric objects (after all, they can be dealt ...
3
votes
0answers
33 views

About an example of normal bundle of a curve over a surface

I know the definition of the normal bundle $N_{C/S}$ of a curve $C$ over a surface $S$ as the cokernel of the injection $T_C \subset T_S|_C$ where $T$ is the tangent bundle. I would like to exhibit an ...