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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Are trivial vector bundles on curves semistable?

Let $C$ be an irreducible projective curve with at worst nodal singularities. Let $E$ be the trivial locally free sheaf of rank $r$ i.e., $E$ is the direct sum of $r$ copies of the trivial line ...
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1answer
42 views

Calculating intersection number of $(x^2+y^2)^3-4x^2y^2=0 $ and $x=0$ at $(0,0)$

I am trying to find the intersection number of $(x^2+y^2)^3-4x^2y^2=0 $ and $x=0$ at $(0,0)$. The intersection number of $F$ and $G$ is defined to be $dim_k(O_p/(F,G))$(Here $O_p$ is the local ring ...
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1answer
51 views

Is the finite union of algebraic curves an algebraic curve? [closed]

Is the finite union of algebraic curves an algebraic curve? I'm kind of new to the study of algebraic curves and I believe this is intuitive.
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2answers
42 views

Is it true that for every $n\in\mathbb{N}$ there exists an algebraic curve $C$ and a point $p$ in that curve whose tangent plane has dimension $n$?

The title is very self-explanatory, I was thinking of finding a curve $k^n$ (k is the field with a non-singular point p, such that its tangent plane is V(0) and thus would be $k^{n}$ whose dimension ...
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1answer
39 views

On projection $\mathbb{P}_A^n \rightarrow \mathbb{P}_A^{n-1}$

I am learning about rational maps at the moment, and the notes I am reading gives an example, the projection $\mathbb{P}_A^n \rightarrow \mathbb{P}_A^{n-1}$ given by $[x_0, ..., x_n] \rightarrow ...
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1answer
86 views

on rational normal curves and determinantal representation

Background: Let \begin{align} \Omega = \begin{bmatrix} L_1 & L_2 & \cdots & L_n \\ M_1 & M_2 & \cdots & M_n \end{bmatrix}\end{align} be a matrix of linear forms on ...
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1answer
68 views

General Structure Sheaf Question

In reading the Red Book of Varieties and Schemes, I am confused on this idea boxed in the image: . Why does $F(x)=0$ for all $x \in U$ imply $F=0$? Can't we only say this if $F$ vanishes on all of ...
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37 views

is $Hom(P,N \otimes_{End(P)} P) = N$?

This is probably well known to people who work with algebras but I couldn't find a reference. Say I have a ring A and a module P and I take B = End(P), the endomorphism ring. Let N be a B-module, is ...
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1answer
49 views

Questions on branch points on elliptic curve

So let $(E,p)$ be an elliptic curve over a field $k$ with a choice of $k$-valued point $p$. Then by Riemann-Roch, there are two global sections of $\mathcal{O}_{E}(2p)$ which gives a double cover of ...
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37 views

“Projective tangent space” to a projective variety

Is there an established notation for the linear subvariety tangent to a projective variety $V$ at a point $x$? I've seen this called the "projective tangent space" in some places. The closest thing ...
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1answer
55 views

Groupoid of $G$ torsors over spectrum of a finite field: some clarifications

I am trying to read Barghav Bhatt's online notes which show that if $f : X \to Y$ is a morphism of varieties over a finite field $\mathbb {F}_q$, then the generating function counting the image of the ...
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1answer
39 views

What is an ideal residue, and the square of an ideal?

In Zariski's "The Concept of a Simple Point of an Abstract Algebraic Variety", in section 2.1, he refers to elements being '$m$-residues', where m is the maximal ideal of a quotient ring $p(W) / ...
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39 views

All closed subsets of the image of $\mathbb C^n$ under rational mapping

Let $p_1,\dots,$ $p_{n}$, $q_1,\dots,q_{n}$ be polynomials in the variables $z_1,\dots,z_m$ and $$ Y:=\{(z_1,\dots,z_m):\ q_1(z_1,\dots,z_m)\ne 0,\dots, q_{n}(z_1,\dots,z_m)\ne 0\}\subset \mathbb ...
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1answer
35 views

Definition of a sheaf: What is $s\rvert_{V_i}$ if $V_i\not\subseteq U$?

I am reading Hartshorne's book on algebraic geometry, which defines a sheaf to be a presheaf $\mathscr F$ on a topological space $X$ such that: For all open sets $U$ and open coverings $\{V_i\}$ of ...
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1answer
32 views

Help in this exercise in Fulton's algebraic curves book

I'm trying to solve the exercise 8.37 (page 111) in Fulton's algebraic curves book: I've already solved almost every item, it miss just the equivalence: The curve $X$ has a hyperelliptic ...
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0answers
30 views

Relative tangent space and flatness

Let $C, X$ be a smooth projective variety over an algebraically closed field. Suppose $f:X \to C$ is a flat, proper morphism with geometrically irreducible fibers. Is the relative tangent space ...
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23 views

A way to prove the focal property of the ellipse

Every light ray which is radiant from a focal point reflects on the ellipse, such that it goes through the other focal point. Assuming $P=(x_0,y_0)$ is an arbitrary point of an ellipse with the ...
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0answers
35 views

Simplification of a definition in Hartshorne's algebraic geometry book

I'm reading Hartshorne's book and on page 53 he defines intersection multiplicity of a projective variety and a hypersurface: I'm wondering if we can simplify this definition if we take $Y$ to be a ...
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1answer
28 views

Help in this teminology in Hartshorne's algebraic geometry book

I'm studying Hartshorne's Algebraic Geometry book and on page 51: What the author means by $M_{\mathfrak p}$ and "length"? I suppose $S_{\mathfrak p}$ is the localization of the ring $S$ at ...
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27 views

Invertible Prime in a Noetherian Scheme

What means the following: Let $\ell$ be a prime, and $X$ a Noetherian Scheme, on wich the prime $\ell$ is invertible. The reference is the appendix of "Weil Conjectures, Perverse Sheaves and l’adic ...
3
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1answer
38 views

blow-ups and fields of definition

Let $X$ be a smooth variety over some field $k$, not necessarily algebraically closed. Let $Z$ be a closed subvariety of $X$ which is only defined over an extension of $k$ (e.g. a closed point which ...
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0answers
34 views

What is the order of a cusp form at a cusp?

This question is about the definition of order of a section of a bundle at a point, and the related notion of associated divisor. Let us look at a specific example, the discriminant $\Delta(z)$ on ...
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43 views

Analytic interpretation of the expression ``a property $\mathcal P$ holds for generic $v_1,\dots,v_k\in V$''

I try to use a theorem that states that some property $\mathcal P$ holds for generic $v_1,\dots,v_k\in V$, where $V\subset\mathbb C^n$ is an irreducible variety of dimension $d$. To apply the ...
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17 views

Basis of $L(D)$

Let $L(D)=\{f\in k(C)\mid \text{ord}_P(f)\ge -n_p,\ \text{for all $P$ in C}\}$ be the vector space defined on page 99 of Fulton's algebraic curves book. I would like to know how to find a basis ...
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0answers
55 views

Calculating object position in 3D space

I'm looking for an algorithm to calculate the position of point P in space using a triangular(/rectangular) plane on the 'ground'. The position between the points ABC of the triangle on the ground are ...
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1answer
30 views

Determining irreducible components of $Z(y^4 - x^6, y^3 - x y^2 - y x^3 + x^4) \subset \mathbb A^2$

I'm trying to determine the irreducible components of the zero set $Z(I)$ for the ideal $I = (y^4 - x^6,\, y^3 - x y^2 - y x^3 + x^4)$, in the affine space $\mathbb A^2$ over an algebraically closed ...
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2answers
50 views

Geometric Intuition Behind Blowing Up a Cusp on a Plane Curve?

I'm reading Hartshorne AG V.3 on monoidal transformations and embedded resolutions. I understand one sort of intuition behind blowing up a point on a surface (or more generally a subvariety of a ...
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35 views

short exact sequences of complexes and triangles in the homotopy category

Suppose I start with an abelian category $\mathcal{A}$, form its category of complexes $C(\mathcal{A})$ and consider a short exact sequence in this category: $$0 \to A^{\bullet} \to B^{\bullet} \to ...
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1answer
38 views

Equivalence definitions of hyperelliptic curves

I'm reading Fulton's algebraic curves book and on page 111, he defines hyperelliptic curves. For him an hyperelliptic curve $C$ is a curve which has a hyperelliptic weierstrass point $P$, i.e., $2$ is ...
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1answer
98 views

Calculating global sections of sheaves

Consider the usual projective space $\mathbb{P}^{1} = \mathbb{C} \cup \{\infty\}$, and the Weil divisor $D = \{0\} \subset \mathbb{C} \subset \mathbb{P}^{1}$. Writing projective space as the union of ...
3
votes
1answer
68 views

Translation from schemes to varieties

At the moment, I know very little algebraic geometry (sadly!) so I apologise for the silliness/stupidity of these questions. Set up: Let $k$ be a field (not necessarily algebraically closed). Take ...
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1answer
34 views

How do you compute the pull-back of a complex differential (1,1)-form given its potential?

Let $F: X \to Y$ be a holomorphic map. Let $\omega$ be a complex differential $(1,1)$-form on $Y$, $\omega=\partial \bar \partial f$, where $f$ is a pluri-subharmonic function. How would one ...
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1answer
35 views

Properties of the functor $(X, \mathcal{O}_X) \mapsto (X(k), \mathcal{O}_{X(k)})$

Let $k$ be an algebraically closed field. In Görtz and Wedhorns book one can read about an equivalence of categories $\{\text{integral schemes of finite type over } k\} \to \{\text{prevarieties over ...
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2answers
94 views

Counting the number of solutions of equation $x^2 + y^2 = 1$ over $\Bbb Z/p$

List proofs of the fact that the number of solutions to $x^2 + y^2 = 1$ over $\Bbb Z/p$, where $p$ is a prime $\neq 2$, is $p-(-1)^{\frac{p-1}2}$. I thought of two. I write one below.
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55 views

Isomorphism of divisors

Consider the cartier divisor group $CDiv_{T_{N}}(X_{\Sigma})$ defined on the fan $X_{\Sigma}$. I am having trouble proving the following assertion that there is a natural isomorphism ...
4
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1answer
66 views

Maximal ideals of polynomial ring

We know that if $k$ is algebraically closed, then each maximal ideals of $k[x_1, x_2, \ldots , x_n]$ are of the form $(x_1 - a_1, x_2 - a_2, \ldots, x_n - a_n),$ where $a_1, a_2, \ldots , a_n \in k$ ...
2
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1answer
32 views

quotients by quasi-coherent ideal sheaves

I saw this lemma stated in some lecture notes: If $\mathcal{I}$ is a quasi-coherent sheaf of ideals on a scheme $X$ and if $U$ is any affine open subset of $X$, then ...
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1answer
42 views

Bridgeland stability conditions: The heart satisfies the Harder-Narasimhan property

Given a stability condition $(Z,\mathcal{P})$ on a triangulated category $\mathcal{D}$. Take $\mathcal{A}=\mathcal{P}((0,1])$. Then $\mathcal{A}$ is the heart of a bounded t-structure on ...
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0answers
54 views

Help in a proof in Fulton's algebraic curves book

I'm reading Fulton's algebraic curves book and I didn't understand this proof of proposition 7 (page 106) very well: So I have the following doubts: I didn't understand why $\text{ord}_P(f')\ge ...
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1answer
36 views

Connected component identification?

Suppose I give a random 2 variable polynomial relation such as: $$x^3+y^3=10$$ $$x^2 + 7yx^4 + x^2-15=0$$ Etc... How do I determine how many individual pieces there are to the graph?
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1answer
49 views

function field of $zy^2 - x^3$ in the plane

I am interested in understanding the connection between abstract curves and smooth projective curves. So I looked at a simple example $zy^2 - x^3$ in $\mathbb{P}^2$. The function field can be computed ...
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1answer
34 views

If a curve is hyperelliptic, we have an equality in Clifford's Theorem

I'm studying Fulton's algebraic curves book and I have the following question: Clifford's theorem says that if $D$ is a divisor and $W$ is a canonical divisor with $l(D)\gt 0$ and $l(W-D)\gt 0$, then ...
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0answers
29 views

Complex elliptic surface with 24 $I_1$ fibers

Is a complex elliptic surface with 24 $I_1$ fibers always a K3 surface? Is ti possible to characterize a K3 surface in terms of the singular fibers of a given elliptic surface?
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1answer
40 views

functor of points for grassmannian

I am reading section 16.7 of Vakil's Foundations of AG, which constructs the Grassmannian G(k,n) via its functor of points, namely B -> surjections O^n_B -> Q, where Q is locally free of rank k. He ...
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16 views

Minimize cost function, provide algebraic expression and determine optimal input sequence if $N = 4$

Given the discrete-time linear system $x(t+1) = 2x(t) + u(t)$ with $x(0) = 0$ and $t\geq0$, the goal is to find an optimal input sequence $u^{*}(0), \dots, u^{*}(N-1)$ that minimizes the following ...
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49 views

Visualizing projective closures - is it okay to just think of the affine case?

This question is quite general and has been discussed on MSE before, however my case is a little bit different and I'm wondering about the geometric interpretation of a specific example. I think that ...
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0answers
37 views

Distinguished points of a cone

Sorry, as this is a rather trivial question that I am misunderstanding, but I do not understand how the distinguished point is defined. We define it as a homomorphism from some semigroup $S_{\sigma}$ ...
3
votes
1answer
41 views

Equivalence relation on regular functions

In this problem, consider $K$ an algebraic closed field and $X\subset\mathbb{A}^n_k$ an irreducible variety. Given an open Zariski $U\subset X$, we say that a function $\phi:U\rightarrow K$ is regular ...
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0answers
87 views

properties of pullback diagrams

Suppose you have a commutative diagram: $\require{AMScd}$ $\begin{CD} A @>>> B\\ @VVV @VVV \\ C @>>> D \\ @VVV @VVV \\ E @>>> F \end{CD}$ Let $T$ be the top "square", $B$ ...
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41 views

Jacobian matrix rank and dimension of the image 3

Let $p_1,\dots,p_n,q_1,\dots,q_n$ be polynomials in $m<n$ variables and $f(z)=\left(\frac{p_1(z)}{q_1(z)},\dots,\frac{p_n(z)}{q_n(z)}\right)$. By construction, the variety $V$ is the Zariski ...