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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How to complete Vakil's proof that the composition of projective morphisms are projective when the target is quasicompact?

For this question, a morphism $\pi : X \rightarrow Y$ is projective iff there exists a finite type quasicoherent sheaf $\mathcal{E}$ on $Y$ such that $X$ is isomorphic (as a $Y-$scheme) to a closed ...
2
votes
1answer
27 views

Complementary Text to Gunning and Rossi - Analytic functions in several complex variables

I'm currently a second year student who has a background in group theory, ring theory, galois theory, metric spaces and point set topology. I'm currently taking courses in algebraic topology, advanced ...
0
votes
2answers
27 views

Proof that an affine scheme is quasi compact

I want to prove that any affine scheme $X = \operatorname{Spec} A$ is compact. $\bigcup D(f_i)$ is an open cover of $X$ if and only if the sum of the ideal $\sum (f_i)$ contains 1$. That is $$ D( \...
2
votes
1answer
31 views

How does $Mor(V,W)$ becomes a vector bundle and what are the transition functions of it?

Let $p_1:V\rightarrow X$ and $p_2:W\rightarrow X$ be two algebraic vector bundle over a variety $X$ of rank $n$ and $m$ respectively. Let's say $\{U_i,f_i\}$ and $\{U_i,g_i\}$ are trivializations for ...
2
votes
1answer
7 views

Of $n^2$ points of intersection, $np$ lie on curve of deg. $p < n$, then remaining $n(n - p)$ lie on a curve of deg. $n - p$

Let $C$, $C'$ be two plane curves of degree $n$. Is the following statement true or not? Suppose that of the $n^2$ points of intersection, $np$ lie on a curve of degree $p < n$, then the ...
1
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0answers
12 views

Equivalence of two definitions of proper morphism for separated varieties

In Harthshorne (and pretty much everywhere else I know) a proper morphism is defined to be A morphism of schemes $f:X\to Y$ is said to be proper if it is sperated, of finite type and ...
3
votes
1answer
26 views

Lang-Nishimura theorem still carries through or fails when assumptions are dropped?

Theorem (Lang-Nishimura). Let $X \,-\!\!\rightarrow Y$ be a rational map between $k$-varieties, where $Y$ is proper. If $X$ has a smooth $k$-point, then $Y$ has a $k$-point. Does the theorem of Lang-...
1
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0answers
24 views

Max Noether's Fundamental Theorem proof in Fulton's book

In the end of the proof we get that $H=A'F+B'G$ and $A'=\sum A'_i$, $B'=\sum B'_i $ while $A'_i$ and $B'_i$ are forms of degree $i$. I don't understand how then he makes the conclusion that $H=A'_sF+B'...
4
votes
1answer
42 views

Grothendieck ring of varieties

In the context of Grothendieck ring of varieties there is often used notion of variety over variety (for example here -2.2.1). I always used only varietes over field. My question is : how is it define ...
0
votes
1answer
21 views

How to get all parameters of an east opening hyperbola given some points

I have a symmetric east opening hyperbola. I am only interested in the positive parts and I know the vertex point (a,0). I also know some random point on the curve. I thought this is enough to get all ...
0
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0answers
23 views

Simultaneous row echelon form

I am working on the following problem right now: Let $V,W,H$ be finite dimensional vector spaces. I have a group action $Gl(V)\times Gl(W) \curvearrowright Hom(V\otimes H,W)$ in the obvious way i.e. ...
0
votes
0answers
29 views

Show that the set $\{ P \in E(\mathbb{Q})\ |\ h(P) \le M \}$ is finite, for any constant $M$.

Show that the set $\{ P \in E(\mathbb{Q})\ |\ h(P) \le M \}$ is finite, for any constant $M$. Here $h(P)$ is logarithmic height of $P$, that is, $h(P):=\log H(P)$ and $H(P)=H(x)$, for $P=(x,y) \in E(...
3
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0answers
55 views

Background for reading Atiyah's first paper on the twisted cubic

What should an undergraduate student need to know before being able to read Michael Atiyah's A Note on the Tangents of a Twisted Cubic ? Most of the words in the paper look foreign to me, but I'm ...
1
vote
1answer
24 views

Variety structure on the set of Borel subgroups is noncanonical?

If $B$ is a Borel subgroup of connected linear algebraic group $G$, the fact that all Borel subgroups of $G$ are conjugate, combined with the fact that $N_G(B) = B$, shows that the set of Borel ...
3
votes
0answers
50 views

Sheaf Theory: When do we have $(\mathscr{F}/\mathscr{G})(U) = \mathscr{F}(U)/\mathscr{G}(U)$?

I was originally unsure about this in a more basic situation: if I had a quasi-coherent ideal sheaf $\mathscr{I}$ on a scheme $X$ then on an open affine subscheme $U \subset X, U = Spec(A)$ we have $\...
6
votes
1answer
92 views

example of toric varieties with nontrivial first cohomology group

If I remember correctly, when a toric variety is smooth or simplical (the moment polytope is simplicial rational), then there is no odd dimensional cohomology group and the dimension of the even ...
4
votes
2answers
54 views

Classify all schemes of degree $2$ and $3$ over $\mathbb{R}$ supported at the origin in $\mathbb{A}_\mathbb{R}^2$

Classify all schemes of degree $2$ and $3$ over $\mathbb{R}$ supported at the origin in $\mathbb{A}_\mathbb{R}^2$. In particular, show that while any such scheme $X$ whose complexification $X \times_{\...
4
votes
1answer
49 views

Vector bundle whose sections don't vanish anywhere

Suppose I have a vector bundle $V$ over a smooth projective variety $X$ with the following exotic property: any global section of $V$ does not vanish anywhere. The general question what can we say ...
3
votes
0answers
38 views

Correspondence to Parameter Space, but not Moduli Space?

I've been thinking recently about some nice moduli problems in algebraic geometry as well as the relationship of moduli spaces to string theory, gauge theory, and such. Mathematically, to my ...
0
votes
1answer
190 views

rotating parabolas in 3D to get part of a circle

Say you have a unit-circle with its center at $(0,0)$, and you "cut out" the upper-right quadrant. You rotate this segment around the Y-axis and the orthographic projection is the upper-right segment ...
-1
votes
1answer
21 views

minimization on a u-shape curve

There is a function y=f(x), f'(x)<0, for all values of x. There is another function y=g(x),g'(x)>0, for all values of x. There is a third function y=h(x), which is u-shaped. We assume that both f-h ...
3
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0answers
42 views

Image of $\mathbb{P}^1 \times \mathbb{P}^1$ by Segre embedding is a hyperboloid

Let $f: \mathbb{P}^1 \times \mathbb{P}^1 \to \mathbb{P}^3$ be the Segre embedding given by $((x_0:x_1),(y_0:y_1))\mapsto(x_0y_0,x_0y_1,x_1y_0,x_1y_1)$. Gathmann's notes claim that the real points of ...
1
vote
0answers
46 views

Can you explain Tate's algorithm by the example?

Consider the elliptic surface $\mathcal{E}$ over $\mathbb{P}_k^1$ with homogenous coordinates $t$, $s$ and a field $k$ of even characteristic: $$ \mathcal{E}\!: s^{10}(y^2z + yz^2) = t^4s^6x^3 + (t^...
11
votes
3answers
684 views

Prove $X^2+Y^2-1$ is irreducible using geometrical tools.

I'm trying to understand what is meant in this paragraph: of "Conics and Cubics. A Concrete Introduction to Algebraic Curves (by Robert Byx)": He wants to prove that the polynomial $X^2+Y^2-1$ is ...
3
votes
0answers
64 views

Canonical bundle of the Lagrangian Grassmannian

I'd like to compute the canonical bundle of the Lagrangian Grassmannian $\mathbb{LG}_n$, the set of Lagrangian subspaces of dimension $n$ of a complex vector space together with fixed symplectic ...
2
votes
1answer
50 views

Question about ramified morphism from smooth variety to projective space

Suppose $f:X\rightarrow P^n$ be a $d:1$ ramified covering over the complex $\mathbb{C}$. Here $X$ is smooth projective variety. I have some questions. a) what is direct image of structure sheaf $f_*...
1
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0answers
35 views

Understanding $O_{\mathbb{P}(F)}(d)$ on $\mathbb{P}(F)$ (in local coordinates)

Say $F$ is a vector bundle of rank $r+1$ on some scheme $X$ with transition maps (cocycle) $\psi_{ij}$ (with respect to some open cover $U_i, U_j,\ldots, U_k$ of $X$). We denote by $\mathbb{P}(F)$ the ...
2
votes
1answer
32 views

What does additive mean in “additive basis” in algebraic geometry?

Let $Gr(k,n)$ be the set of all $k$-dimensional subspaces of an $n$-dimensional vector space. I heard that people say "an additive basis" of $\mathbb{C}[Gr(k,n)]$. What does additive mean? Thank you ...
2
votes
0answers
34 views

Geometric equivalent of the degree zero divisor class group of an algebraic function field (in the singular case)

In an algebraic function field $F/k$ we have the degree zero divisor class group $\text{Cl}^0(F/k)$. Now since any such function field corresponds to the function field of a normal projective curve ...
3
votes
1answer
38 views

What happens with negative plurigenus?

It is a well known result that for a smooth, projective k-variety, the dimension of the global section $H^0(X,K_X^j)$ of $j$-powers of the canonical bundle. Also called plurigenus, are birational ...
3
votes
0answers
30 views

Handelman's Theorem on Nonnegative polynomial in Compact Polytope?

Background: Representing polynomials by positive linear functions on compact convex polyhedra by David Handelman Consider a polynomial $f\in K[x_1,\ldots,x_n]$ in a compact polytope, related ...
1
vote
1answer
82 views

Is this regular function globally rational?

Let $\mathbb{k}$ be an algebraically closed field of characteristic not 2 or 3, and let $X \subseteq \mathbb{A}^2_\mathbb{k}$ be the locally closed subset given by $X = \{ (x,y) : x^3=y^2, (x,y) \...
1
vote
4answers
145 views

Survey articles in Commutative/Homological algebra

I am a graduate student interested in Commutative algebra/Homological algebra. I am comfortable with first eight chapters of Atiyah. I am familiar with some algebraic geometry, first two chapters of ...
15
votes
1answer
777 views

Class group of $k[x,y,z,w]/(xy-zw)$

I had a homework problem (II.6.5 in Hartshorne) to compute the (Weil divisor) class group of $X=\operatorname{Spec} k[x,y,z,w]/(xy-zw)$. I have accomplished this; however, I used some results I don't ...
1
vote
2answers
39 views

Why does every irreducible component have codimension one?

I'm a bit confused on the following lemma. Here $\mathbb{P}(V)$ is the set of one dimensional subspaces of $V$. With the choice of a basis for $V$, there is a natural bijection $\mathbb{P}(V) \...
2
votes
0answers
36 views

Noether Normalisation lemma

How do we explain Noether normalisation in terms of line bundles and linear systems? If $X$ is a smooth irreducible projective variety over an algebraically closed field of char 0 of dimension $r$. ...
1
vote
1answer
33 views

Idempotents in coordinate ring

Let $k$ be an infinite field. Let $V$ be an algebraic set of $k^n$. Let's suppose that $f(x)+I(V)\in k[x_1,\cdots,x_n]/I(V)$ $(x=(x_1,\cdots,x_n))$ is such that $f(x)^2+I(V)=f(x)+I(V)$. Then does $f(x)...
4
votes
1answer
55 views

Geometric differences between $\operatorname{Spec}\mathbb{C}[x]/(x^2-x)$ and $\operatorname{Spec}\mathbb{C}[x]/(x^3-x^2)$

As far as I can tell, the topological spaces associated to the schemes in the title are both sets with two elements, with the discrete topology since both have prime ideals $(x)$ and $(x-1)$ which are ...
2
votes
1answer
110 views

Vakil's FOAG, Exercise 9.2.K: Transcendental Complex Numbers

How does one realize a transcendental complex number as a maximal ideal of $\mathbb{Q}(t) \otimes_{\mathbb{Q}} \mathbb{C}$? This is the essence of Exercise 9.2.K in Vakil's FOAG. Here is what I've ...
4
votes
2answers
204 views

Hartshorne generically finite morphisms (II, 3.7)

I have a question concerning one of the exercises of Hartshorne, Ch. II. Namely: Exercise 3.7 about gerneically finite morphisms. A morphism $f: X \rightarrow Y$ with Y irreducible and $\eta$ ...
1
vote
1answer
56 views

Basic question on isomorphism between section of structure sheaf of an affine scheme

Let $A$ be a commutative ring with unity. In Ravi Vakil's notes he defines $O_{Spec A} (D(f))$ to be the localization of $A$ at the multiplicative set of all functions that do not vanish outside of $V(...
6
votes
2answers
172 views

What's the difference between $\mathbb{A}^n$ and $\mathbb{A}^{n+1}$?

Besides the obvious difference in topological dimension. If you want to distinguish between $\mathbb{R}$ and $\mathbb{R}^2$, take an open set in the plane, remove a point, then it's still connected. ...
1
vote
1answer
13 views

Nichtnegativstellensatz the same as Handelman's Theorem?

Wikipedia on "Handelman's theorem: If $K$ is a compact polytope in Euclidean $d$-space, defined by linear inequalities $g_i ≥ 0$, and if $f$ is a polynomial in $d$ variables that is positive on $K$, ...
0
votes
0answers
15 views

Polynomial in Compact Polytope: Algebraic Description for the Compact Polytope?

Consider a polynomial $f\in K[x_1,\ldots, x_n]$ where $K=\mathbb R$. For example $$f[x_1,x_2,x_3]=x_1 x_2+x_3$$ \begin{eqnarray*} x_{1} & \in & [0.2,0.5]\\ x_{2} & \in & [0,1]\\ x_{3}...
5
votes
2answers
102 views

Geometric interpretation of primitive element theorem?

The primitive element theorem is a basic result about field extensions. I was wondering whether there are nice geometric ways to visualize it or think about it. Since field spectra are singletons, it ...
5
votes
2answers
85 views

How can I compute a presentation of the tangent bundle for a smooth manifold defined by a family of polynomials?

Consider a smooth manifold $M$ given by a system of smooth functions $$ \begin{align*} f_1 = 0 \\ \cdots \\ f_k = 0 \end{align*} $$ in $n$ variables. This has the algebraic description as the $\mathbb{...
1
vote
2answers
61 views

Understanding Conics in Pencil

In a paper I'm reading about ellipses they talk a lot about "pencils of conics", after looking around on the web to learn more like this website: http://planetmath.org/pencilofconics I found some ...
2
votes
2answers
234 views

Upper semicontinuity of fibre dimension on the target

This is Vakil 18.1.C. Suppose $\pi : X \to Y$ is a projective morphism where $Y$ is locally Noetherian (or more generally $\mathcal{O}_Y$ is coherent over itself). Show that $\{y \in Y : \dim \pi^{-1}(...
2
votes
1answer
26 views

Can I choose $k+1$ hypersurfaces to avoid a fiber of dimension $k$ in projective space?

Let $X$ be a closed subscheme of dimension $k$ in $\mathbb{P}^n_A$, where $A$ is a Noetherian ring. In Exercise 11.3.C of Ravi Vakil's notes, it is shown that one may choose $k+1$ hypersurfaces such ...
0
votes
1answer
22 views

Sections of a finite étale cover of a connected scheme which coincide at a geometric point

Let $\phi_1, \phi_2 : S \longrightarrow X$ be two sections of a finite étale cover $X \longrightarrow S$ of a connected scheme $S$. Assume that $\phi_1 \circ \overline{s} = \phi_2 \circ \overline{s}$ ...