The study of geometric objects defined by polynomial equations. Algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc.

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100 views

Kähler differentials of a hyperelliptic curve.

Let $f:X \rightarrow \mathbb{P}^1_k$ be a hyperelliptic curve, where k is a field of characteristic not 2. $\mathbb{P}^1_k$ is the union of $U = Spec k[t]$ and $V = Spec k[s]$ where $s= 1/t$. This ...
3
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1answer
68 views

Krull's intersection theorem in the q-expansion principle

I'm currently reading the proof of the q-expansion principle in Katz'73 paper "p-adic properties of modular schemes and modular forms" . The principle itself is a Corollary (1.6.2) of Theorem 1.6.1, ...
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1answer
110 views

quasi-affine/projective varieties | f=g on dense subset | diagonal subset | how to show that (f,g) is continuous?

Let $f: X \to Y$ and $g: X \to Y$ be morphisms in the category $(QProj-k)$ (its objects are quasi-projective and quasi-affine $k$-varieties). Show that $f=g$ if and only if $f$ and $g$ are identical ...
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156 views

Hartshorne, Chapter 1, Projective varieties, Question 4(b)

Chapter 1, section 2, question 4(b) in Algebraic Geometry says An algebraic set $Y \subseteq \mathbb{P}^n$ is irreducible iff $I(Y)$ is a prime ideal. I'm confused about the solution given in ...
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184 views

How to see that tangent cones really are cones?

I'm currently following an introductory course on algebraic geometry. When studying projective varieties, which we defined as $V(f)=\{ [x] | f[x]=0 \}$ given that f is a homogeneous polynomial, the ...
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139 views

Ramification divisor and Hurwitz formula of higher dimensionanl vaireities

Assume $X,Y$ are smooth varieties, $f: X \to Y$ is a separated morphism. Then it is claimed that there is Hurwitz formula: $$K_X \sim f^*K_Y + R$$ with $R$ an effective diviosr. I try to prove this ...
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69 views

Where can I find this result?

A Noetherian scheme $X$ over an algebraically closed field $k$ the set of derivations $\mathcal{O}_{X,x} \to \kappa(x)=k$, is isomorphic to the Zariski tangent space $(\mathfrak{m}/\mathfrak{m}^2)^*$ ...
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1answer
37 views

Is the map $(g_\ast \mathcal{O}_Y)_{g(p)} \to \mathcal{O}_{Y,p}$ always injective

Let $\psi : A \to B$ be a ring homo and let $g : \mbox{Spec} B \to \mbox{Spec} A$ be induced map. Is true that the map $(g_\ast \mathcal{O}_Y)_{g(p)} \to \mathcal{O}_{Y,p}$ is injective? $Y = ...
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94 views

Morphisms between quasiprojective varieties preserve irreducibility

Let $X,Y$ be two quasiprojective varieties and $\phi \colon X \to Y$ a surjective morphism. Let $Z \subset Y$ a closed set such that $\phi^{-1}(Z)$ is irreducible. Prove that $Z$ is irreducible. ...
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109 views

If a divisor $D$ on a surface has positive self-intersection, then $nD$ or $-nD$ has nontrivial sections, eventually in $n$.

Let $S$ be a smooth complex projective surface. Let $D$ be a divisor on $S$, such that $D^2 = D.D> 0$. Then, at least one of the following holds: For $ n \gg 0$, $H^0(nD) \neq 0$; for $n \gg 0$, ...
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1answer
113 views

What is the stalk at a point of the quotient of a scheme?

This came up when I was talking to @Benjalim on the chat. Consider the space $X=\operatorname{Spec} \mathbb C[x]$. With the usual structure sheaf, this is a scheme. Let $Y$ be $X$ with the points ...
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1answer
156 views

Elementary questions on ample invertible sheaves

Let $X$ be a quasi-compact scheme. Let $\mathcal L$ be an invertible sheaf on $X$. We say $\mathcal L$ is ample if for any finitely generated quasi-coherent sheaf $\mathcal F$ on $X$, there exists ...
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83 views

Question regarding Hartshorne Example II.(6.5.2)

Let $k$ be a field, let $A=k[x,y,z]/\langle xy-z^2\rangle$ and let $X=\operatorname{Spec}A$. Let $Y:y=z=0$ I want to know the divisor of $y$ In Hartshorne book, because $y=0 \Rightarrow z^2=0$ and ...
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48 views

Hartshorne III.7.6b) (ii) => (i) "Duality for a projective scheme)

Let X be a closed immersion of dimension n in P = *P*$^N_k$, where k is an algebraically closed field. Let $\omega_P$ denote the canonical bundle and A the local ring $\mathcal O_{P,x}$. Then ...
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1answer
65 views

How to imagine a plane defined by Cartesian Plane Equation?

It isn't difficult for me to imagine a plane based on three points. Also it is quite simple to imagine a plane based on point and normal vector. Are there some tricks to imagine a plane defined by ...
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129 views

Transition functions of trivial vector bundle

I have a question on trivial vector bundles. The question is as follows: Can we characterize the transition functions of a trivial vector bundle in some way? To be very concrete: suppose we ...
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1answer
106 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 ...
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1answer
57 views

All local cohomology modules being zero

Let $R$ be a Noetherian ring with unit, $I$ be an ideal of $R$ and let $M$ be a finitely generated $R$-module. Suppose $H_{I}^j(M)=0$ for all $j$, then how can one show that $M=IM$? The converse of ...
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60 views

square system of polynomial equations having infinite number of solutions

Suppose we have a system of $n$ polynomial equations in $n$ unknowns over $\mathbb{C}$ and suppose that the corresponding ideal generated by these equations is not the unit ideal $(1)$. Under what ...
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83 views

All the Associated Primes are minimal.

Let $R$ be a commutative Noetherian ring with unit and let $I$ be a fixed ideal. I am sorry if the following turns out to be a very silly question. 1) Suppose $\operatorname{Ass}(R/I)$ are all ...
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1answer
83 views

Computing a rational function at a point in terms of a uniformising parameter

I am not quite sure how to ask this precisely, but vaguely I would like to know how difficult it is to write a function on an algebraic curve at a point $P$ as a power series of a uniformising ...
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1answer
204 views

A question on torsion sheaves

Im not sure if Ive got this right: Let X be an integral scheme and $\mathcal{F}$ a coherent sheaf. Then $\mathcal{F}$ is torsion if and only if it is not supported at the generic point. It is is easy ...
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2answers
81 views

Showing the equations of 3 lines are dependent

I have 3 lines: $$ A_1x + B_1y + C_1 = 0 $$ $$ A_2x + B_2y + C_2 = 0 $$ $$ A_3x + B_3y + C_3 = 0 $$ That I believe are dependent. That is, all of the intersections of any pair of two of these lines ...
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84 views

Fiber product an difference of schemes

Let $j:W\to X$ be a closed immersion of a schemes and $f:Y\to X$ a morphism. The basechange of $j$ along $f$ defines a closed immersion $W\times_X Y\to Y$. The inclusion $k:X\setminus W\to X$ and ...
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93 views

Linear subvarieties of $\Bbb{P}^n$ and change of coordinates

I am looking at problem 4.11 of Fulton's Algebraic Curves. He asks to show that if $V = V(H_1,\ldots, H_r)$ where $H_i's$ are all linear forms of degree $1$, then there is a projective change of ...
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1answer
52 views

What is the area of a pixellated surface?

Recently after playing hours of Minecraft (and generating block volumes), a question popped up in my head: Is it possible to easily (by hand in a reasonable amount of time) determine the surface area ...
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1answer
78 views

A scheme from a quasi-compact morphism

Let $f: X \to Y$ be quasi-compact, set $\mathcal{J} = \ker f^{\#}$, and $Z = V(\ker f^{\#}) = \{ y \in Y \: | \: \mathcal{J}_y \neq \mathcal{O}_{Y,y} \}$. $Z$ is a locally ringed topological space ...
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62 views

faithful flatness for the group scheme

Let $K$ is an algebraic closed field, $$f:S{\longrightarrow}T$$ be a map of closed set in $K^n$. Moreover, assume that $f(S)$ is dense in $T$. Show $f(S)$ contains an open dense subset of $T$. It is ...
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1answer
75 views

Formula for index of $\Gamma_1(n)$ in SL$_2(\mathbf Z)$

Is there a precise formula for the index of the congruence group $\Gamma_1(n)$ in SL$_2(\mathbf Z)$? I couldn't find it in Diamond and Shurman, and neither could I find an explicit formula with a ...
3
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1answer
245 views

SVD and how to get two points on a 3D line from the representation of the line by means of two intersecting planes?

I have a 3D line represented by the intersection of these two planes $a_1x+b_1y+c_1z+d_1=0$ $a_2x+b_2y+c_2z+d_2=0$ I need to compute two 3D points $P_1=(x_1,y_1,z_1)$ and $P_2=(x_2,y_2,z_2)$ ...
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1answer
81 views

Hartog's theorem and affineness?

Is the following true? I see it is true for X affine, but I do mot know how to show it otherwise. Let X be a normal noetherian local scheme of dimension 2, with closed point s. Show that $X \setminus ...
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1answer
94 views

Topology of manifolds

Where can I find a stricter presentation of topology of manifolds, then in section 0.4 in Griffiths-Harris? For example, they define the map $H_k \times H_{n-k}$ by presenting a cycle by a submanifold ...
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1answer
87 views

Two questions on projective schemes

Let $n\geq 1$ and $K$ a field. Let $X=Spec(K[X_0,\ldots,X_n]/I)$ be a closed subscheme of $\mathbb{A}^{n+1}$. Let $Y=X\times_{\mathbb{A}^{n+1}}(\mathbb{A}^{n+1}\setminus\{0\})$ denote the closed ...
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1answer
109 views

Easy to state high-dimensional consequences of Bezout theorem

A classical consequence of Bezout's theorem for plane curves is Pascal's theorem. I am curious if there are some other statements that you find pretty that can be formulated (almost) as elementarily ...
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1answer
204 views

Coordinate ring of a cartesian product

I am considering the coordinate ring $k[X \times\mathbb{A}^n]$, where $X$ is an algebraic variety in $\mathbb{A}^n$. I want an isomorphism between this and the polynomial ring $k[X][y_1,\ldots, y_n]$. ...
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1answer
156 views

Irreducible homogeneous polynomial over $\mathbb{Z}$ staying irreducible modulo large primes

Given an irreducible homogeneous polynomial $F \in \mathbb{Z}[x_1,\cdots,x_n]$ (which can be assumed to be irreducible over $\bar{\mathbb{Q}}$ if necessary), Is it true that there exists ...
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1answer
120 views

Intersection of two $n$-dimensional quadratic inequalities?

I have two quadratic inequalities of the form $$ a_1x^TAx + b_1^Tx + c_1 \le 0\\ a_2x^TAx + b_2^Tx + c_2 \le 0 $$ where $A\in\mathbb{R}^{n\times n}$ is positive semidefinite, $x\in\mathbb{R}^n$, ...
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1answer
92 views

Showing that two quadratic surface are normal

In the book "Algebraic Geometry" by Robin Hartshone, or GTM 52 for short, there is a problem of showing that two quadratic surface $Q_1: xy=zw$ and $Q_2: xy=z^2$ in $\mathbb{P}^3$ are normal. I have ...
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1answer
149 views

Algebraic variety as a union of nonsingular subvarieties

Definitions Let $M$ be algebraic variety and let $I$ be the defining ideal of $M$, that is $$I(M) = \{ f \in K[X_1,...,X_n] \mid \forall x \in M : f(x)=0 \}$$ Let $f_1,...,f_m$ be the generators of ...
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1answer
81 views

Compactification problem.

How do I compactify the curve $Q(x,y)=0$ in $\mathbb{P}^1\times\mathbb{P}^1$ where $Q$ is a polynomial ?
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1answer
91 views

Definition of differential of map (in algebraic geometry)

Given two schemes $X, Y$ with a map $f: X \rightarrow Y$, one should have a map $df: T^*Y \times_Y X \rightarrow T^*X$. How is this map defined (using the language of algebraic geometry, rather than ...
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1answer
82 views

Union of categories and $\ell$-adic local systems

I have a question about the definition of $\ell$-adic local systems. I understand how to define local systems over any finite extension of $\mathbb{Q}_{\ell}$, but not how to take the "union" of these ...
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1answer
70 views

What is the tangent space for intersection point of irreducible varieties.

Given a (real) plane, I know the definition of tangent space for irreducible varieities. (If $P \in V(f_1, \cdots, f_n)$ irreducible, then $T_P V = V(f^{(1)}_{1,P},\cdots, f^{(1)}_{n,P})$. Suppose ...
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1answer
97 views

Closed morphism between schemes of finite type over a field induces a closed map between varieties?

Let $X$(resp. $Y$) be a scheme of finite type over a field $k$. Let $f\colon X \rightarrow Y$ be a closed morphism. Let $X_0$(resp. $Y_0$) be the set of closed points of $X$(resp. $Y$). Then $f$ ...
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1answer
128 views

Proof that a Zariski closed subset of a projective space is the common zeros of homogeneous polynomials

Let $K$ be an algebraically closed field. Let $n \ge 0$ be an integer. We consider $K^{n+1}$ as a topological space with Zariski topology. Let $G = K^*$ be the multiplicative group of $K$. Let $X = ...
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1answer
91 views

What is the fiber of an etale map over a point?

Given a field $K$, $K$-schemes $X$ and $Y$, an étale map $f:X\to Y$, $x\in X$ a point of the topological space to $X$ and $y:=f(x)\in Y$ a point of the topological space to $Y$. What is the ...
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1answer
99 views

Algebraic definition of blow-ups

Let $X$ be a scheme. Choose $C\subset X$ be a subscheme of $X$ and let $\mathcal{I}\subset \mathcal{O}$ be the corresponding ideal sheaf. Then $\mathcal{B}=\oplus_{d\ge0}\mathcal{I}^d$ is a sheaf of ...
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1answer
144 views

About first Chern class and Poincaré duality in case of an ample divisor

Let $D$ be a very ample divisor in $X$ projective variety. I can't understand why the first chern class $c_1(\mathscr{O}_X(D))$ equals the Poincarè dual of D, $\mathscr{P}(D)$
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161 views

Discrete Valuation Rings

Let $V = \mathbb A^1(k)$ ($k$ is an algebraically closed field), $\Gamma(V) = k[X]$ and let $K = k(V) = k(X)$. Prove that for each $a \in k = V$, $\mathcal{O}_a(V) := \{f\in K(V): f$ is defined at ...
3
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1answer
217 views

Tensor products commute with inductive limit

How to prove, that tensor products commute with direct limits, if the main ring is not the same? For every $i$ we have modules $L_i$ and $M_i$ over a ring $A_i$, and for every $i \geq j$ homomorphisms ...