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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Why this linear system doesn't have base points?

I see somewhere that linear system of a non-negative degree divisor over a rational curve doesn't have base points, but I didn't understand why. I don't understand what the degree has to do with base ...
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1answer
15 views

Checking a complete linear system on a curve is base point free

I have a vague idea that I can check if a complete linear system |D| on a curve is base point free by comparing $h^0(D)$ and $h^0(D-P)$ for all points P on the curve. Intuitively, I guess the idea is ...
2
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0answers
24 views

Weighted Projective Spaces and varieties

Consider the weighted projective space $\mathbb{P}(1,1,2)$ with variables $x_0,x_1,x_2$ of degrees 1,1,2 respectively. Consider the map $$(a_0,a_1,a_2) \to (a_0^2,a_0 a_1,a_1^2,a_2)$$ with ...
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0answers
16 views

Linear system without base points

Let $D$ be a divisor and $V$ a vectorial subspace of $L(D)$. The set of the divisors $S=\{(f)+D\mid f\in V\}$ is called a linear system. A point $P\in C$ is a base point if for every divisor $D\in S$, ...
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0answers
25 views

Does every irreducible projective cubic curve have a nonsingular point of inflection?

Does every irreducible projective cubic curve necessarily have a nonsingular point of inflection? I've been trying to construct counterexamples, to no avail, which leads me to believe the question can ...
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2answers
21 views

If $ Y $ is irreducible set so is $cl(Y)$. [duplicate]

If $ Y $ is irreducible set so is $cl(Y)$. If $cl(Y)$ is reducible then $cl(Y)= A \cup B$ where both $A$ and $B$ is closed in $cl(Y)$. Now how do we proceed?
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0answers
7 views

How to calculate the horizontal offset of the top Bezier point of an Arc

Given the following: A circle with a diameter D 3 Bezier points P0 P1 and P2 that make an equilateral triangle, and the upper point P1 is at the top of the circle. The distance between P0 and P2 is ...
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2answers
51 views

Hilbert polynomial of twisted cubic 'by hand'?

I am asked to calculate the Hilbert polynomial of the twisted cubic curve \begin{equation*} C = \{(s^3 : s^2t : st^2 : t^3); (s:t) \in \mathbb{P}^1 \} \subset \mathbb{P}^3 \end{equation*} and I know ...
1
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1answer
12 views

Automorphisms of manifold vs automorphisms of Hodge structure

Let $X$ be a compact Kahler manifold. Let $H^k = H^k_{prim} (X)$ be primitive cohomology of $X$ considered as an object in category of of polarized Hodge structures. We have a map \begin{equation*} ...
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1answer
40 views

Exercise about an algebraic surface

Let $\mathbb{P}^6$ the six-dimensional complex projective space. Suppose that $Q_{i}$ is a smooth quadric in $\mathbb{P}^6$ for $i=1,...,4$. Define $$S=Q_1 \cap Q_2 \cap Q_3 \cap Q_4 $$ as the ...
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1answer
46 views

Irreducible projective cubic, exists continuous surjection?

Let $C$ be an irreducible projective cubic in $\mathbb{P}_2$ with a singular point $p$. So consider $f: \mathbb{P}_1 \to C$ defined as follows. Identify $\mathbb{P}_1$ with the set of lines in ...
2
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1answer
42 views

Is a smooth ring extension of a UFD a UFD?

Let $A \subseteq B$ be noetherian integral domains, $A$ a UFD, and $B$ a smooth $A$-algebra (=the definition of a smooth algebra can be found in ...
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77 views

Review on Riemannian Geometry

I'm currently reading through Griffiths and Harris Principles of Algebraic Geometry, and the only subject in the foundational material section that I am not completely comfortable with is riemannian ...
3
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1answer
54 views

Necessarily a homeomorphism?

Let $D$ be the projective curve defined by $y^2z = x^3.$ Consider the map $f: \mathbb{P}_1 \to D$ defined by$$f[s, t] = [s^2t, s^3, t^3].$$Is it necessarily a homeomorphism? Any help would be greatly ...
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0answers
42 views

The degree-genus formula cannot be applied to singular curves in $\mathbb{P}_2$?

(The degree-genus formula) The Euler number $\chi$ and genus $g$ of a nonsingular projective curve of degree $d$ in $\mathbb{P}_2$ are given by$$\chi = d(3-d)$$and$$g = {1\over2}(d-1)(d-2).$$ My ...
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0answers
24 views

Pole Order of Weierstrass Coordinates

I'm trying to understand a proof in Silverman's The Arithmetic of Elliptic Curves. Background: For an elliptic curve $E$, $x, y \in K(E)$ such that $\phi = [x, y, 1]: E \rightarrow C$ is a basepoint ...
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0answers
46 views

A morphism which is not a comorphism of a regular map

In the lecture, we dealt with morphisms, comorphisms and regular maps. The professor then brought the following example: Let $U$ and $V$ be quasi-affine sets over $\mathbb{C}$ and let $\psi \colon ...
6
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2answers
137 views

Why is so difficult to find beginner books in Algebraic Geometry

I don't understand why is so difficult to find a really beginner book in Algebraic Geometry. Let's take for example the Fulton's algebraic curves: An introduction to Algebraic Geometry. I've already ...
2
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2answers
63 views

Elliptic curve $y^2= x^3 + x$ over the finite field $\mathbb{F}_p$ with $p \geq 3$.

Consider the elliptic curve $$E: y^2= x^3 + x$$ over the finite field $\mathbb{F}_p$ with $p \geq 3$. I want to show that $|E(\mathbb{F}_p)| \equiv 0 \mod 4$. I know that, if $p \equiv 3\mod 4$, ...
1
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1answer
31 views

Rational map from affine cone to projective scheme

I've been working through Vakil's MATH 216 notes and have ran into a wall when he discusses the affine cone of a projective scheme in section 8.2.12. Namely, if S is a finitely generated graded ring, ...
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0answers
30 views

Algebraic surfaces in the language of scheme

Are there materials(lecture notes, books...) that deal with algebraic surfaces in the language of schemes? I am not good at/familiar with the analytic way, and also prefer the scheme-theoretic ...
0
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0answers
52 views

Intersection between a hyperplane and a convex polytope [on hold]

We work over $\mathbb{R}^N_+$, where $N \ge 2$. We are facing a situation in which we need to find the intersection between a hyperplane and a convex polytope. In detail, let $V$ be the set of ...
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15 views

Birational equivalent and isomorphic representation of a subalgebra

here I am again with another exercise which gives me a hard time. Let $A$ be the subalgebra of $\mathbb{C}[t]$ of all polynomials $f(t)$ such that $f(1) = f(-1)$. Let X be an alebraic set such ...
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1answer
70 views

The elliptic curve $y^2 = x^3 + 2015x - 2015$ over $\mathbb{Q}$

Consider the elliptic curve \begin{equation*} E: y^2 = x^3 + 2015x - 2015~\text{over}~\mathbb{Q}. \end{equation*} I want to prove that $|E(\mathbb{F}_7)| = 12$, that $|E(\mathbb{F}_{19})| = 19$ and ...
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0answers
30 views

Algebraic approach to Local Analytic Complex Geometry

I'm attending a second course in Complex Analysis from a geometrical point of view. In the final part of the course we have discussed about germs of complex analytic sets and their algebraic ...
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0answers
24 views

What is a projective rational function in Fulton's book?

I'm reading Fulton's algebraic curves and I don't know if I understood correctly this definition on the page 46: What does the author mean by $f,g\in \Gamma_h(V)$ being of the same degree? Since ...
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0answers
24 views

Fulton, algebraic curves exercise 4.11

I'm doing the exercises of the book Fulton Algebraic curves and I'm stucked in the following problem: A subset $V\subset\mathbb{P}^n(k)$ is a linear subvariety of $\mathbb{P}^n(k)$ if ...
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0answers
28 views

Subvarieties in projective spaces

I was doing the problems from Fulton of Algebraic Geometry, studying for my exam and I got stuck in this exercise. Describe all subvarieties in $\mathbb{P}^1$ and $\mathbb{P}^2$. Any help will be ...
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3answers
83 views

Compute the arithmetic genus of the union of the three coordinate axes $Z(x_1x_2,x_1x_3,x_2x_3) \subseteq \mathbb{P}^3$

Compute the arithmetic genus of the union of the three coordinate axes $Z(x_1x_2,x_1x_3,x_2x_3) \subseteq \mathbb{P}^3$ The arithmetic genus of $X \subseteq \mathbb{P}^n$ is ...
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0answers
22 views

Derived direct image of emdedding of projective varieties

Let $i:Y\to X$ be an embedding map between projective varieties. What is the example of $X$ and $Y$ such that the functor $Ri_*:D^b(\text{Coh}\,Y)\to D^b(\text{Coh}\,X)$ is not a fully faithful ...
2
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1answer
32 views

Why this set is the pole set of $z$?

Suppose $V\subset \mathbb P$ a projective variety, $z$ a rational function on $V$ and let's define $J_z=\{F\in k[X_1,\ldots, K_{n+1}\mid \overline Fz\in \Gamma_h(V)\}$ and $S_z$ the polo set of $z$, ...
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0answers
51 views

What's the most general geometry branch?

What is the most general geometry of curves and surfaces? For example, at curves, we define in differential geometry the tangent vector as the derivative of a regular curve, but visually many other ...
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1answer
33 views

Characterisation of closed subschemes of projective spaces

I'm trying to understand a step in the proof of corollary 5.16 in Hartshorne. The statement Let $ A $ be a ring. if $ Y$ is a closed subscheme of $\Bbb{P}_{A}^{r} $ then there is a homogeneous ...
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0answers
25 views

How to make 4 (parametrized) points (in the complex plane) concentric?

I consider a functional equation (see the earlier discussion at MO) $$ f(f(x))= f(x)^2 + x \qquad \text{where also} f(0)=0$$ In the following I write it in a more concise form (with $z$ instead of ...
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1answer
43 views

Describe the normalization of the cusp.

Show that the normalization of $A = k[x_1,x_2] / (x_2^2 - x_1^3)$ is isomorphic to $k[x]$ and describe (for $k$ algebraically closed) the induced map $Spec(k[x]) \to Spec(A)$ I know that $A$ is a non ...
0
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1answer
35 views

Show that this map has not the going-down property.

Let $A= k[x_1,x_2,y] / (x_2^2-x_1^2(x_1+1))$ and $Spec(A) \to Spec(k[x_1,x_2,y])$ the natural inclusion induced by the projection $k[x_1,x_2,y] \to A$. Consider the map $f : Spec(k[x,y]) \to Spec(A)$ ...
1
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1answer
42 views

Endomorphisms in an exact sequence of vector bundles

Let X be a smooth projective variety over $\mathbb{C}$. And suppose we have an exact sequence of vector bundles over $X$. $\qquad\qquad\qquad\qquad\qquad 0\longrightarrow A\longrightarrow ...
3
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0answers
17 views

What is $HC_0(\operatorname{Spec} k[x,y]/(xy))$?

Does anybody know how to compute $HC_0(\operatorname{Spec} k[x,y]/(xy))$? Here $HC_0(-)$ is the zeroth cyclic homology group. I'm curious since $\operatorname{Spec} k[x,y]/(xy)$ can be viewed as the ...
0
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1answer
39 views

Intersection of ample and effective divisors

I believe it is something silly, but I'm a newbie, so why on a surface the intersection of an effective divisor and a divisor from ample bundle is non-negative? In fact, I need that an intersection of ...
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0answers
25 views

Making sense out of the definition for “morphism of geometric spaces”

I'm trying to read "Introduction to Algebraic Geometry and Algebraic Groups" by Demezure and Gabriel and I'm already stuck on the following definition. A geometric space is defined to be a pair $(X, ...
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1answer
41 views

Quick question: could someone clarify me the notion of “$k$-points” of a scheme?

Let $X$ be a scheme, and $k$ be a field. What does it mean by $X(k)$? The reason I am asking this is that there seems to be several definition(or convention) on it, and I can't see why they lead to ...
3
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1answer
34 views

$\overline{\mathbb{Q}}$-valued points of a $\mathbb{C}$-scheme

I've just read the definition of a $T$-valued point on an $S$-scheme from Ravi Vakil's notes (i.e. the morphisms $T \to S$), and something about the definition doesn't seem to make sense intuitively. ...
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0answers
55 views

Examples of vector bundles

I know three examples of interesting vector bundles (not taking into consideration what one gets from them using algebraic operations): Tangent bundle of a smooth manifold Tautological bundle over ...
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26 views

$H^1$ of some vector bundle on a cubic 3-fold

This question is a sequel to the following one Dimension of moduli space of some stable vector bundles on a cubic 3-fold. Let $E$ be a stable rank 2 vector bundle on a cubic 3-fold, say $X$, with ...
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1answer
35 views

Let $k$ be an algebraically closed field and $Y \subset \mathbf{A}^n$ be the set $\{(t,t^2,t^3|t \in k\}$. What is are the gnerators of $I(Y)$

I'm having a problem thinking through this rigorously. Let $k$ be an algebraically closed field and $Y \subset \mathbf{A}^n$ be the set $\{(t,t^2,t^3|t \in k\}$. What is are the generators of $I(Y)$? ...
2
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1answer
44 views

I need help in this proof of this exercise from Fulton's book

I'm reading Fulton's algebraic curves book. I'm trying to understand this solution which I found online of the question 4.17 on page 97. What I didn't understand is why $V(J_z)$ are exactly those ...
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1answer
48 views

If $Y$ is a quasi-affine variety, then dim$Y$ = dim$\overline{Y}$

Reading through the proof of proposition 1.10 in Hartshorne's Algebraic Geometry I found some of it to be unnecessary. Is the following proof correct or can you point out my flawed logic. Let $Z_0 ...
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0answers
41 views

Example 2.3.4 in Hartshorne algebraic geometry

I am reading Hartshorne algebraic geometry. Example 2.3.4. Let $k$ be an algebraically closed field, and consider the affine plane over $k$, defined as $A^2_k = \text {Spec} k[x,y]$ . The closed ...
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0answers
16 views

Classifying all $E$-torsors up to isomorphism?

If $0\to E\to F\xrightarrow{\phi}\mathbb{A}^1_X\to 0$ is an exact sequence of bundles over the scheme $X$, then it is known that if $s\colon X\to \mathbb{A}^1_X$ is the constant section sending ...
2
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1answer
61 views
+50

Classification of $3$-pointed rational curves

I tried to prove that $\mathbb P^1 \setminus \{0,1,\infty\}$ is the fine moduli space for the moduli problem, which assigns to a scheme $S$ the set of (isomorphim classes of) $4$-pointed rational ...