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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1answer
24 views

Simple points of an algebraic variety from an analytic point of view

I am a specialist in fuctional analysis, but from time to time I have to use some results from algebraic geometry, and every time I face great difficulties in translating them into the language ...
2
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1answer
25 views

Deriving the Quadratic Polynomials Defining the Twisted Cubic

I've recently been reading about rational normal curves and how they may be represented and have come to the following question: (for simplicity's sake, the problem is stated in terms of the RNC in ...
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24 views

Extensions of algebraic varieties

Fix an algebraically closed field of characteristic $0$ and let $X$ be the variety corresponding to the ideal $I_X$ of $k[x_1,\ldots, x_n]$ generated by $$A_X=\{f_i(x_1,\ldots,x_n)\vert ...
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19 views

For which points of circle function $\frac{1-y}{x}$ is regular [duplicate]

Consider circle $X = V(x^2+y^2-1)$ and rational function $\varphi = \frac{1-y}{x}$ on $X$. I want to know for which points $\varphi$ is regular(in sense of algebraic geometry). Intuitively i think ...
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0answers
30 views

Noether's Normalization Lemma

In trying to understand the attached proof of Noether's Normalization lemma, I am having trouble with the induction step. How do we know that the transcendence degree of S is still n? Thanks in ...
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23 views

Proof that $\text{span} \{v_1,…,v_k\} \cap \text{ker}(T) = \{0\}$ if $\{v_1,…,v_k\}$ are vectors in general linear position.

The problem set up is as follows: Let $\omega^{(i)} \in \mathbb{R}^n$, for $i=1,2,...,k$, $k \le n$, be i.i.d. random vectors (whose distribution is irrelevant). Also, let $A \in \mathbb{R}^{m \times ...
1
vote
1answer
34 views

Reference in EGA

I was reading the proof of a statement in EGA 0.10.3.1.3 (hope I got that right, it's in the third volume actually), which I am in need of. There are several references to other parts of EGA, ...
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37 views

The intersection multiplicity of two coprime polynomials is less or equal than the multiplicity of their product?

Are given $H_1,H_2$ coprime polynomial of $K[X,Y]$ with $K$ a algebraic closed field, $P\in\mathbb{P}^2(K)$ a point. Holds that$$\mu_P(H_1,H_2)\leq m_P(H_1H_2)$$ where $\mu$ is the intersection ...
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0answers
9 views

Computing the order of a divisor in the Jacobian of a hyperelliptic curve.

Given a hyperelliptic curve of genus $g$, of equation $H: y^{2}+h(x)y=f(x)$ and defined over the finite field $\mathbb{K}$, how does one compute the order of a (reduced) divisor defined over ...
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30 views

Coproduct of projective schemes

In EGA II 5.5.6 it is said that the coproduct of two projective schemes is projective. I don't understand why this is true. It is mentionned that it is a consequence of the definition of a projective ...
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12 views

Open embedding of affine toric varieties implies Cone is face of the other

let $\tau, \sigma \subseteq N_{\mathbb{R}}$ be two rational, strongly convex polyhedral cones with $\tau \subseteq \sigma$. Now we get an inclusion $S_{\sigma} \to S_{\tau}$ inducing an inclusion ...
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1answer
33 views

Explicit equations of an image and its normality

One has a mapping $\phi:\mathbb{A}^2\longrightarrow \mathbb{A}^4$ given by $\phi(x,y)=(x,xy,y(y-1),y^2(y-1))$. First, I am trying to find explicit equations defining $\mathrm{Im}(\phi)$. It is easy to ...
0
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1answer
40 views

Subset of points in noetherian scheme of rank $\le n$ is open

Let $\mathcal{F}$ be a coherent sheaf over a Noetherian scheme $X$. Lets define its rank in a point $x \in X$ as the dimension of $\mathcal{F}_x \otimes k(x)$ (here $k(x)$ is the residue field in ...
0
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1answer
42 views

Extending of morphism from $Spec \ \mathcal{O}_{X,x}$

Let $X$ be an integral scheme over $S$ and let $Y$ be a scheme of finite type over $S$. Let $x \in X$. How can I show that a morphism from $Spec \ \mathcal{O}_{X,x}$ to $Y$ can be extended to a ...
3
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1answer
42 views

On a proof of the fact “A projective nonsingular curve minus a finite number of points is affine”

In This notse, Vakil gives a proof of the theorem stated in the title. (page 5). In the proof he made use of a section s of $\mathcal{O}_C(kp)$ that has only one zero of order k at p. However such a ...
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0answers
27 views

Computing the cotangent complex: what's the ring?

As far as I understand, deformation theory of schemes may be calculated via the cotangent complex. I have read that in general the cotangent complex may be difficult to compute. However, I have a ...
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0answers
35 views

Determining a scheme $X$ is affine from $Qcoh(X)$

My question is a subquestion of this question. I do not want to use full reconstruction theorems. The settings is the following. Let $X$ be a scheme. Assume that the adjunction in TAG01BH of the ...
0
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0answers
52 views

resolution of a surface singularity

Let $f: Y \to (0\in X)$ be a resolution of a surface singularity and $R^1f_*\mathcal{O}_Y=0$. Then $H^1(Y,\mathcal{O}_Y)=0$. Why? Is it clear? How can I show it?
3
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2answers
69 views

Any hyperelliptic curve is never a complete intersection.

Show that any hyperelliptic curve is never a complete intersection. As any curve of genus greater than 1 is either hyperelliptic or canonical, I think we can equivalently show that any curve of genus ...
0
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1answer
24 views

Reducible cubic surface are always singular.

I want to prove that Any reducible cubic surface are always singular. A possible way may be to take a look at the intersection of the irreducible components. But I don't know how. Thanks for any ...
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1answer
35 views

Closed immersions and complete linear systems

Let $X$ be a local complete intersection subscheme in $\mathbb{p}^n$ for some integer $n>0$. Denote by $i:X \to \mathbb{P}^n$ the induced closed immersion, ...
2
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1answer
46 views

Any curve of genus three is either hyperelliptic or trigonal?

A curve $C$ is said to be trigonal if it admits a rational function $f: C \to \mathbb{CP}^1$ of degree $3$. Is any nonsingular plane curve of degree four trigonal? Can the map be chosen so that it has ...
2
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1answer
50 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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29 views

Examples of base points of linear systems

I'm reading Fulton's algebraic curves book and we have the following definitions: A divisor $D=n_1P_1+\ldots,n_kP_k$ ($n_i$'s are integers and $P_i$'s are points) over a curve. A linear system as ...
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1answer
27 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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22 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 ...
4
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0answers
39 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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1answer
45 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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26 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
53 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 $\epsilon$, ...
1
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1answer
26 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 ...
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1answer
53 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
vote
1answer
29 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 ...
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0answers
40 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
30 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
21 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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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 ...
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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 ...
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2answers
60 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 ...
3
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2answers
48 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
66 views
+50

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
51 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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87 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
56 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
44 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 ...
2
votes
1answer
38 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 ...
1
vote
1answer
59 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 ...
9
votes
2answers
182 views

Why is it so difficult to find beginner books in Algebraic Geometry?

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