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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2
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19 views

Divisors of differentials.

Let $\mathbb{C}$ be the base field. Suppose $C \subset \mathbb{P}_2$ is a nonsingular projective curve of degree $d > 3$. Must it be the case that $D \in \text{Div}(C)$ is the divisor of a ...
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1answer
16 views

Lines on a quadric surface

I want to show that: Let $Q$ be a quadric surface of rank 3 in $\mathbf{P}^3$(over $\mathbf{C}$), then any line in $Q$ must pass through the only singular point of $Q$. I know that up to a transform ...
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0answers
11 views

if $D_1\sim D_1'$ and $D_2\sim D_2'$, then we have $D_1+D_2\sim D_1'+D_2'$

I'm studying Fulton's algebraic curves book and I'm trying to prove that if $D_1\sim D_1'$ and $D_2\sim D_2'$, then we have $D_1+D_2\sim D_1'+D_2'$. If the the first two equivalences work, then we ...
3
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0answers
25 views

maximal linear subspaces contained in the cone over the Clifford torus.

Forgot: this is about Find a subspace of $\mathbb{R}^4$ for which $x^T*A*x$ = 0 I was a little surprised to find that, in the cone $x^2 + y^2 = z^2 + w^2$ in $\mathbb R^4,$ there are infinitely many ...
2
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0answers
16 views

Irreducible curve contained in linear subspace

Can someone give me a starting point for the following question? I don't know where to begin! Let $C \subset \mathbb{P}^n$ be an irreducible curve of degree $d$. Show that $C$ is contained in a ...
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0answers
24 views

lattices and torsion free sublatices.

I have the following statement that I cant proof, which according to my book is trivial. Let $N$ be a lattice. Let $N_1 \subset N$ be a sublattice such that $N/N_1$ is torsion free. Then it followes ...
3
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1answer
48 views

If a polynomial maps a region onto a neighborhood of zero, does it follow that it has a zero in some “robust” sense?

Let $B^n\subseteq\Bbb R^n$ be a unit ball, $P: B^n\to\Bbb R^m$ is polynomial in each component, and assume that the image of $P$ contains $0$ in its interior. Does it follow that for some $\epsilon$, ...
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1answer
21 views

Is a map from a smooth curve, bijective away from the singularities, already a normalization?

Over $\mathbb{C}$, I am considering a map between projective curves $f: C \rightarrow C'$, where $C$ is smooth. Suppose that $f$ is surjective as well as bijective away from the singularities of ...
0
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1answer
38 views

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 ...
1
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1answer
26 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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38 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 ...
0
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0answers
22 views

Genus formula for a curve in a $2$-dimensional complex torus?

For a curve $C$ in a $2$-dimensional complex torus $T$, is there any formula to compute the genus of $C$? Say, in terms of self-intersection number? For a curve in $\mathbb{P}^2$ or a K3 surface, ...
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0answers
20 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
36 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 ...
0
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2answers
23 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 ...
0
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2answers
55 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
18 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*} ...
0
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1answer
47 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 ...
7
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1answer
47 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
votes
1answer
50 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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0answers
80 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
55 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
43 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
27 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
49 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
votes
2answers
146 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
votes
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
35 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
31 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
57 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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0answers
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 ...
3
votes
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
29 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 ...
3
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3answers
86 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 ...
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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
52 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 ...
0
votes
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 ...
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1answer
38 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
vote
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
votes
0answers
18 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
votes
1answer
40 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 ...
0
votes
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, ...
5
votes
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 ...