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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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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14 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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8 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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16 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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18 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 ...
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36 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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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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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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28 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 ...
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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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16 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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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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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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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!
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85 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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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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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 $spec (k [x, x^{-1}])$. ...
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61 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
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 ...
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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 ...
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2answers
63 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
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 ...
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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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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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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 ...
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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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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
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 ...
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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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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} = ...
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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) ...
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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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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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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 ?
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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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30 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
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?
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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 ...
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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
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 ...
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1answer
96 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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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 ...
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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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2answers
63 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 ...
2
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2answers
69 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 ...