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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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
48 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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9 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!
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72 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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20 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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21 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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31 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 $spec (k [x, x^{-1}])$. ...
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57 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 ...
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3answers
62 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 ...
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21 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 ...
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2answers
51 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 ...
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1answer
41 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 ...
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40 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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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 ...
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23 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 ...
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2answers
23 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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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 ...
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1answer
57 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 ...
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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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25 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} = ...
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44 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) ...
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25 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 ...
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39 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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35 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 ?
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1answer
65 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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25 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 ...
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1answer
43 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?
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35 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 ...
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38 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 ...
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1answer
62 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 ...
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1answer
89 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 ...
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1answer
29 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 = ...
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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 $s_i$ ...
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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 ...
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31 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 ...
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0answers
18 views

Compactifying affine algebraic families

Suppose I have a smooth morphism $f:X\to S$ of affine varieties over an algebraically closed field of arbitrary characteristic. I want to regard this as a family of varieties parametrized by $S$ and ...
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1answer
44 views

Proof that $H^{0,1} \oplus H^{1,0} = H_{DR}^1$

I am struggling with a proof from Donaldson's Riemann Surfaces which he leaves as an exercise. we want to construct an isomorphism from the direct sum of $H^{1,0}(X)$, the set of holomorphic 1-forms ...
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2answers
68 views

Sum of Betti numbers of a degree $d$ hypersurface of $\mathbb{P}^n_{\mathbb{C}}$

Let $X \subseteq \mathbb{P}^n_{\mathbb{C}}$ be the zero set of a homogeneous polynomial of degree $d$. Assume that $X$ is smooth. Is there a way to find the sum of the Betti numbers of $X$ (i.e. the ...
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1answer
39 views

Riemann-Roch analysis of point divisor ring on smooth genus 3 Riemann surface

Let $C=C_4\subset\mathbb{P}^2$ be the smooth genus 3 Riemann surface given by a quartic curve. Let $P\in C$ be a point, and $D=P$ the divisor given by the point $P$. Let ...
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8 views

How can a nonconvex polytope be defined (not by an LMI)?

A convex polytope can be defined by an LMI (linear matrix inequality) or a list of points. How can a nonconvex polytope be defined?
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27 views

Maximum volume inscribed ellipsoid inside nonconvex polyhedron

In Convex Optimization (Boyd, Vandenberghe), an algorithm for finding the maximum volume inscribed ellipsoid inside a convex polyhedron is given on p. 414 (8.4.2 Maximum volume inscribed ellipsoid) [I ...
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37 views

Is there affine surface of general type of the form $y^2=f(x) f(z)$ or $y^2=f(x) g(z)$?

Let $f,g$ be univariate polynomials with integer coefficients of degree $n$. Is there affine surface of general type of the form (1) $y^2=f(x) f(z)$ or (2) $y^2=f(x) g(z)$? I would expect for $n$ ...
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1answer
47 views

Questions about algebraic curve definition

The algebraic curve definition states as following $S_{2}^m$ denotes homogeneous polynomial of degree $m$ in $x$ and $y$ $f_{m}(x, y) = \sum_{j, k \ge 0, j+k=m} C_{j,k}x^{j}y^{k}$ The elements of ...
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1answer
69 views

Every commutative ring is a quotient of a normal ring?

In the book Étale cohomology by Milne I found on p. 37 (in the context of constructing the henselization of a local ring) the following claim: "Every ring is a quotient of a normal ring". The same is ...
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1answer
37 views

What is the intersection of the Segre variety in $\mathbb{P}^5$ and the Veronese surface in $\mathbb{P}^5$?

This is an exercise from Chapter 8 of Ideals, Varieties and Algorithms by Cox et al. The projective Veronese surface in $\mathbb{P}^5$ is defined as the projective closure of the surface $S$ which ...
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0answers
53 views

Proofs of Hodge duality: $H^{0,1}(X) = H^{1,0}(X)^*$

I am looking for a proof of this fact, where $H^{1,0} = Ker(d: \mathscr{E}^{1,0} \rightarrow \mathscr{E}^{2})$ and $H^{0,1} = Coker(\overline{\partial}: \mathscr{E} \rightarrow \mathscr{E}^{0,1}$, ...
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1answer
72 views

Complementary textbook algebraic geometry

I don't know where to ask this or if it is allowed to do it, so please let me know any details for further questions of this kind. I am taking an algebraic geometry class and am using the textbook ...
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0answers
38 views

Complexification of proper scheme

Let $X$ be a proper scheme over $\mathbb{C}$. We define $X_{\mathbb{R}}$ to be a scheme over $\mathbb{R}$, which is the same topological space as $X$ with structure sheaf generated by real and ...
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2answers
50 views

Finding parabola parameter given 2 points

How can I determine which is the directrix and the focus of a parabola and what is the distance between those points, only knowing that this parabola has its symmetry axis = OX and its passes through ...
4
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0answers
45 views

Proof of Serre duality for $D=0$

I have been working through a proof of Serre duality, which proceeds by induction on the divisor $D$, but I am having trouble with the base-case. How can I prove that on a riemann surface X, $H^0(X, ...