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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46 views

Geometric statement of Prime Avoidance?

The Prime Avoidance Theorem is very clean to state in algebraic terms: Let $I \subset R$ be an ideal (with $R$ noetherian) and $I \subseteq \bigcup_{i=1}^r P_i$, where each $P_i$ is prime. Then $I ...
1
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1answer
32 views

Zero set of a homogeneous element of degree $0$, or how $D_+(2)\subset \text{Proj}(\mathbb{Z}[x])$ looks like.

Let $S=\bigoplus_{n=0}^\infty S_n$ be a graded ring. We denote $S_+=\bigoplus_{n>0}^\infty S_n$. As usual we define $\text{Proj}(S)$ to be the set of homogeneous, prime ideals $\mathfrak p$ of $S$ ...
0
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2answers
68 views

Proving that certain subset of the Grassmannian is open in the Zariski topology.

Let $\mathbb{G}(k,n)$ be the Grassmannian of $k$-planes in $\mathbb{P}^{n}$, and let $X\subseteq\mathbb{P}^{n}$ be an irreducible algebraic variety. Fix a positive integer $m$. We define $$ ...
3
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0answers
62 views

sections of higher direct image sheaf

Let $f:X \rightarrow Y$, be a proper birational morphism of projective algebraic varieties with $X$ smooth. Denote by $R^if_* \mathcal{O}_X$, the higher direct image sheaves. Do exists a simple way ...
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0answers
23 views

Looking for a formula to map a 2d pixel coordinate to a region within a grid.

I am given a pixel bounding box of the form: (x1, y1), (x2, y2) Where (x1, y1) is the bottom left coordinate and (x2, y2) is the top right coordinate of the ...
2
votes
2answers
84 views

When regularity of $A$ implies regularity of $A[w]$?

Let $A$ be a commutative noetherian ring (I do not mind to assume that $A$ is a UFD), and assume that $A$ is regular. Recall that a commutative noetherian ring is called regular if all its ...
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0answers
16 views

Points of Weil restriction

Let $l/k$ be a finite separable extension of fields. Let $X$ be an $k$-scheme such that the Weil restriction $Y:= R_l/k(X_l)$ exists, where $X_l$ is the base-change of $X$ to $l$. By definition of the ...
2
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0answers
30 views

Hartshorne Exercise 2.6: what gradation does $S(Y)_{x_i}$ inherit from $S$?

Let $S=k[x_0,\dots,x_n]$ be the "homogeneous polynomial ring" of $\mathbb P^n$ and let $S(Y)_{x_i}$ denote the localization at the image of $x_i\in S(Y)$ of $S(Y)=S/I(Y)$. In Hartshorne, he asks us to ...
1
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1answer
16 views

Linear system which gives $(m,n)$-polarization?

What is the dimension of $H^0(T,\mathcal{L})$, where $T$ is a complex torus of dimension $2$ and $\mathcal{L}$ is a line bundle which gives $T$ a $(m,n)$-polarization?
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1answer
32 views

A curve of genus $g\geq 2$ has a closed point of degree at most $2g-2$ over base field.

I am working on the following problem [R. Vakil] Exercise 19.8.B: Suppose $C$ is a curve of genus $g>1$ over a field $k$ that is not algebraically closed. Show that $C$ has a closed point ...
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0answers
34 views

Intersection of affine open subschemes

If X is separated scheme, then for any affine opens U, V, the intersection is affine. If X is quasi-separated, then the intersection can be covered by finitely many affine opens. Is there some ...
2
votes
1answer
38 views

Explicit description of the inverse image sheaf of an ideal sheaf.

$\DeclareMathOperator{\Spec}{Spec}$ Let $f: \Spec A \to \Spec B$ be a morphism of affine schemes and $f^\#: B \to A$ be the corresponding ring homomorphism. Let $\mathcal{I} \subseteq ...
0
votes
1answer
30 views

$div(z)=0\Leftrightarrow z\in k$

I'm reading algebraic curves book from Fulton and I didn't understand this corollary on page 98: Why $\deg(div(z-\lambda_0))\gt 0$? and why is this a contradiction? Thanks a lot
2
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0answers
32 views

Number of parameter of a quadric

Suppose for example that $S$ is an algebraic complex surface contained in $\mathbb{P}^6$. $S$ is the complete intersection of four quadrics in the six dimensional projective space. If i take a quadric ...
1
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1answer
23 views

What is precise definition of $monomial~curve$ in affine e-space?

What is precise definition of $monomial~curve$ in affine e-space ?
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1answer
22 views

Equivalence of points on a smooth cubic in $\mathbb{P}^2$

Let $C \subset \mathbb{P}^2 (\mathbb{C})$ be a smooth cubic. Show that $(p)$~$(q)$ if and only if $p=q$. $p$ and $q$ are points, and two divisors $D$ and $D'$ are ~ if $D-D' = (f)$ for some ...
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0answers
40 views

Fano surfaces all of whose rational points lie on some geometric line

Are there any ? Namely let $X$ be a smooth del Pezzo surface defined over $\mathbb{Q}$ that has rational points and such that the degree of the del Pezzo is small, say $d=3$ or $4$. Is it possible ...
1
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1answer
83 views

Hilbert Series of $\mathbb{C}^2$?

Consider the following ideal in the polynomial ring $\mathbb{C} [x,y,z]$: \begin{equation} I = \langle z^2, yz \rangle \end{equation} One can compute the Hilbert series of the affine varieties ...
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0answers
38 views

Visualizing Curve in Projective Space

I'm wondering if anyone can help me understand/visualize what the curves xy=1 and $y=x^3$ look like in projective space. I'm familiar with the construction of the space, and how to homogenize the ...
1
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1answer
40 views

Automorphisms of non-hyperelliptic curve of genus 3 in $\mathbb{P}^{2}$

I have a question from R. Vakil's exercise 19.7.C which goes as follows: Suppose $C'\subset\mathbb{P}^{2}$ is a smooth plane quartic curve. Show that there is bijection between automorphisms of $C'$ ...
6
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0answers
57 views

Finding equations for projective curves, low genus, Riemann-Roch.

Let $C \subset \mathbb{CP}^n$ be a nonsingular projective curve, and let $L \subset \mathbb{CP}^n$ be a hyperplane. We have that $L \cdot C$ is a divisor $H$ on $C$ if $C \subset L$. Let $R = ...
2
votes
0answers
35 views

Exercise on Fulton's “Algebraic Curves”

Exercise 7.12 from Fulton's Algebraic Curves Find a quadratic transformation of $\; F = Y^2 Z^2 − X^4 −Y^4$ with only ordinary multiple points. By checking the partial derivatives, I found that ...
3
votes
1answer
49 views

Real Lie groups and elliptic curves

Let $f:A\to A'$ be a morphism of elliptic curves over the real numbers $\mathbb R$. It canonically induces a morphism $f(\mathbb R): A(\mathbb R)\to A'(\mathbb R)$ between the sets of real points, ...
4
votes
1answer
56 views

Finding generators of toric ideals

Consider the affine toric variety $V \subset k^{5}$ parametrized by $$\Phi(s,t,u) = (s^{4},t^{4},u^{4},s^{8}u,t^{12}u^{3}) \in k^{5}$$ where k is an algebraically closed field of characteristic 2. ...
2
votes
1answer
43 views

Why the kernel of the restriction map $\text{Pic}^0(X)\to \text{Pic}(U)$ is finitely generated?

Let $X$ be a smooth and geometrically connected projective curve over a field $k$ and $U$ be a non-empty open subset of $X$. Let $\text{Pic}^0(X)$ be the degree $0$ Picard group of $X$ and ...
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0answers
39 views

Better understanding regular functions on a Projective variety

Hi guys I was just looking an example from class that was left as obvious, but it is not so obvious to me. $W= V(x_1x_4-x_2x_3)= $ where $I(W)= \langle x_1x_4-x_2x_3 \rangle$ so we just picked an ...
2
votes
0answers
43 views

What is meant by the discriminant locus of a fibration?

If we have a fibration $f: X \to B$ (allowing singular fibers) of a differentiable manifold $X$, precisely what is meant by "the discriminant locus of $f$ " $\Delta \subset B$ and how do we define it? ...
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0answers
37 views

Computing singular locus

I should compute the singular locus of $V(x^n+y^k)$ where $k,n\ge 1$ are natural numbers and $f=x^n+y^k\in K[x,y]$. Here $K$ is a field of characteristic $p$ (which means, as far as I understood it, ...
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2answers
47 views

Smoothness of $A \subseteq C$ implies smoothness of $B \subseteq C$? where $A\subseteq B \subseteq C$

Let $A \subseteq B \subseteq C$ be commutative rings (noetherian integral domains, if this helps). Assume $C$ is a smooth $A$-algebra. Is it true that $C$ is a smooth $B$-algebra?
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0answers
21 views

Centers of divisorial valuations on toric varieties

Suppose we are given a divisorial valuation $\mathcal{v}$ on a smooth toric variety $X_{\Sigma}$ (for a fan $\Sigma$), i.e. $\mathcal{v}$ is the valuation induced by a torus invariant divisor ...
1
vote
1answer
51 views

Ideal sheaf is aquasi-coherent sheaf of ideals.

Let $X$ be a scheme, For any closed subscheme $Y$ of $X$, the corresponding ideal $I_Y$ given by the kernel of the morphism $i^{\#}:\mathcal{O}_x\rightarrow i_{*}\mathcal{O}_y$ is quasi-coherent sheaf ...
1
vote
1answer
38 views

Varieties and subsets

I'm studying for my last exam and I got stuck in this exercise, from Fulton of Algebraic Geometry. Let $V$ be an affine variety, $f\in \Gamma(V)$ a) Prove that $V(f) = \{P\in V | f(P)=0\}$ is a ...
2
votes
0answers
38 views

Intersection of nef divisors

Let $X$ be a projective variety over $\mathbf{C}$ of dimension $n$. Let $D_1,\cdots,D_n$ be nef Cartier divisors on $X$. Is it true that $(D_1 \cdots D_n)^n \geq D_1^n \cdots D_n^n$? I know that ...
0
votes
0answers
40 views

pushforward/pullback map of sheaves

Let $f:X \rightarrow Y$ be a morphism of schemes. Let $F$ be a sheaf on $Y.$ When is it true that the natural map $F \rightarrow f_* f^* F$ is an isomorphism?
3
votes
1answer
112 views

What is $\mathbb{P}^1-\{0,1,\infty\}$ and why it is interesting?

I found it in this note http://swc.math.arizona.edu/aws/1998/98BuiumLN.pdf Of course, it is the projective line minus three points. But I don't believe it is as simple as it looks like. What can we ...
1
vote
1answer
39 views

If $C \subseteq \mathbb{P}^2$ is a plane curve, then $genus(C)=\frac{1}{2}(d-1)(d-2)$. Compare with example in the notes

In my Algebraic Geometry notes (see http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf) there is the following exercise: If $C \subseteq \mathbb{P}^2$ is a plane curve of degree ...
2
votes
1answer
41 views

Formula for top self intersection of exceptional divisor

Let $X$ be a projective variety over $\mathbf{C}$ of dimension $n$. Let $\pi: Y \to X$ be the blow-up of a smooth point $x \in X$. Is there a nice formula for the intersection number $E^n$?
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0answers
11 views

Is there a pluricanonical divisor on a relatively minimal complex elliptic surface that can be written as sum of fibres?

A complex algebraic surface $S$ is said to be elliptic if there are a smooth curve $B$ and a surjective morphism $p \colon S \to B$ whose generic fibre is an elliptic curve (i.e. a smooth curve of ...
2
votes
0answers
23 views

regular function defined on $V(F)$ without using closed map of morphisms on projective varieties.

Let $k$ be an algebraically closed field and let $F\in k[x,y]$ be an irreducible polynomial. I want to prove that if $z$ is a function regular on all the points of $X=V(F)$ then $z$ is constant. I ...
1
vote
1answer
20 views

homotopy module of a simplicial module

I'm reading a paper about the cotangent complex and I'm having trouble with one of the definitions (3.4 of http://homepages.math.uic.edu/~bshipley/iyengar.pdf ). Let $V$ be a simplicial $R$-module. ...
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votes
2answers
22 views

Finding Coefficient given 2 Equations of Lines and an Angle

A line has equation $$3x - ky = 0$$ Find the value of k if this line makes an angle of 45 degrees with the line $$2x + 5y - 17 = 0$$ The answer among the choices is supposed to be $7$. But I keep ...
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votes
1answer
75 views

Exercise $1.8$ of chapter one in Hartshorne.

In exercise 1.8 of chap I in Hartshorne algebraic geometry, Let $Y$ be an affine variety of dimension $r$ in $\mathbf A^n$. Let $H$ be a hypersurface in $\mathbf A^n$, and assume that $Y ...
8
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0answers
91 views

Intuitive/geometric way of thinking about effective divisors?

What is the motivation/intuition/geometric way of thinking about an effective divisor? I know that a divisor is effective if all its coefficients are non-negative. We write $D \ge 0$ for ...
3
votes
1answer
39 views

Step in the construction of the global spec of a sheaf of algebras

I'm working my way through the construction of the global spec of a sheaf of algebras. Here is the setup. Let $ Y $ be a scheme. Let $ \mathscr{A } $ be a quasi coherent sheaf of $ ...
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votes
0answers
27 views

What is the definition of Osculating plane in algebraic geometry?

I'm studying Fulton's algebraic curves book and in order to understand this paper in Algebraic Curves I need the definition of the d-dimensional osculation plane. Can I understand properly this ...
2
votes
1answer
75 views

Simple question about extending morphisms to $\mathbb{P}^1$

A trivial question, but my lack of working experience in algebraic geometry is a hurdle. Show that every morphism from $\mathbb{A}^1-\{0\}$ to $\mathbb{P}^1$ extends to a morphism from ...
2
votes
0answers
19 views

Question regarding example of toric variety and generators of cone

Consider the canonical example taking n=2, and taking the cone $\sigma$ generated by the vectors $e_{2}$ and $2e_{1} - e_{2}$. The dual cone $\sigma^{v}$ is defined as the set of vectors in the dual ...
0
votes
0answers
25 views

Properties preserved by fppf morphisms

Which properties P do fppf morphisms preserve? In other words if $f: X \to Y$ is fppf and $Y$ has P, for which P does $X$ also have P? I'm particularly interested in the cases when P=smooth or ...
3
votes
0answers
113 views

Moduli space of algebraic surfaces Vs moduli space of curves

Define the surface $S$ as the complete intersection of four quadrics $Q_i$ with $i=1,2,3,4$ in $\mathbb{P}^6$ (complex six dimensional projective space) i.e. $$S=Q_1 \cap Q_2 \cap Q_3 \cap Q_4$$. Put ...
2
votes
1answer
48 views

Equivalence of line bundles and $\mathbb{G}_m$-torsors

This appears to be a duplicate of (half of) this question, but it received no attention so I'll try again. Given a line bundle $L\to X$ on a scheme $X$ over a field $k$, I am to show that ...