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. ...

learn more… | top users | synonyms (1)

-1
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
votes
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 $ ...
0
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
114 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 ...
0
votes
0answers
56 views

What is the geometric interpretation of a P-primary component of an affine k algebra?

Let $R= K[x_1, x_2,...,x_n]$ for some algebraically closed field K. If $I \subset R$ is an ideal, and P is a prime minimal over I, I know that $Z(P)$ is a maximal irreducible subset of $Z(I)$. But ...
4
votes
1answer
45 views

Blow-up of pair of intersecting lines

I have the reducible variety $X=\mathbb{V}(x_1x_2)\subset\mathbb{A}^2$, which is a pair of lines intersecting transversely, and I would like to compute the blow-up at the origin. The Rees ring of the ...
1
vote
1answer
31 views

Isomorphic projective subvarieties, non-isomorphic rings

If $S \subset \mathbb{P}^n$ is a closed set (in the Zariski sense) then $\mathcal{I}(S) \subset k[x_0,\ldots,x_n]$ denotes the homogeneous ideal of polynomials which vanish at $S$. I want to find an ...
2
votes
1answer
79 views

A categorical approach to algebraic geometry

I learned Algebraic Geometry in a geometrical viewpoint, e.g. Hartshorne's book. But I want to learn algebraic geometry categorically, for examples, i) Sheafification $\mathcal{F}^+$ of a presheaf ...
3
votes
0answers
41 views

Properties of polynomials that are polynomial conditions on the coefficients

There are many occasions where we can check whether a (set of) polynomial(s) $f_i$ satisfies certain properties, simply by evaluating a fixed polynomial on the coefficients of the $f_i$. Many times, ...
1
vote
0answers
35 views

Does a field of transcendence degree n correspond to a variety?

There's an equivalence of categories between the category of (nonsingular projective) curves over a field $K$ (with dominant morphisms) and finitely generated fields $L/K$ of transcendence degree 1 ...
2
votes
0answers
45 views

Birational morphism $\mathbb{P}^n \to \mathbb{P}^n$

Let $f: \mathbb{P}_{\mathbb{C}}^n \to \mathbb{P}^n_{\mathbb{C}}$ be a birational morphism. Question: Is $f$ necessarily an isomorphism? I know that if $f$ is in addition to the assumptions above ...
2
votes
0answers
33 views

Relation between the number of halfspaces and the number of vertices of a convex polytope

Suppose we have an $n$-dimensional convex polytope $\mathbf{P}$ represented by an intersection of half-spaces as the following: \begin{equation} \mathbf{P} = \{ x \in \mathbb{R}^n \mid x \in ...
3
votes
0answers
41 views

Kähler differentials, define valuation?

See my previous question for a definition of the $K$-module of Kähler differential $\Omega_{K/k}$. This question is sort of a follow up on it. Suppose $k$ is a field of characteristic $0$, $R$ is a ...
2
votes
0answers
57 views

Kähler differentials, define valuation?

See here for a definition of the $R$-module of Kähler differential $\Omega_{R/k}$. Suppose $k$ is a field of characteristic $0$, $R$ is a $k$-algebra, and let $K$ be a finite extension of $k(x)$. If ...
0
votes
0answers
18 views

Result showing that a certain valuation ring in some function field has to be a DVR?

I know that if $R$ is a valuation ring such that $0 \to \mathbb{C} \to R$ is a left-split exact sequence, then there exists a discrete valuation ring $C$ with $R \subset C$ so that $0 \to \mathbb{C} ...
0
votes
0answers
16 views

Find least-square equation of ellipsoidal cylinder from a set of point

I work in mechanical engineering , and I made a 3D- measure of a drilled surface. SO now I have a set of cartesian coordinates(x,y,z) of the surface and I know the surface has the shape of a cyclinder ...
1
vote
1answer
39 views

A problem while deriving the equation of an ellipse

While deriving the equation of an ellipse, let $0 < k < a$, $(x, y) \in R ^ 2$ (1): $\sqrt{(x + k) ^ 2 + y ^ 2} + \sqrt{(x - k) ^ 2 + y ^ 2} = 2a$ (2): $\sqrt{(x + k) ^ 2 + y ^ 2} = 2a - ...
2
votes
1answer
37 views

Isomorphic function fields of projective curves, bijection of points.

Suppose curves $C$, $D \in \mathbb{CP}^2$ are nonsingular. If their function fields are isomorphic, i.e. $K_C \cong K_D$, then do we necessarily have a bijection of points on $C$ and $D$? Can we do ...
0
votes
0answers
50 views

Kodaira dimensions

I am learning about schemes, specifically abstrac varieties over $\mathbb{C}$ and I want to read a good book or reference where explain the kodaira dimensions, somebody can help me?
3
votes
3answers
107 views

Exercise 1.9 in Hartshorne - is my initial attempt a good start?

Hartshorne's Chapter 1, exercise 1.9 asks us to show that irred. components of $Z(\mathfrak a)$ have dimension $\geq n-r$ if $\mathfrak a$ is an ideal generated by $r$ elements. I think I've reduced ...
6
votes
0answers
80 views

Calculation with Leray spectral sequence

The Leray spectral sequence is a cohomological spectral sequence of the form $$H^p(Y;R^q f_*(F)) \Longrightarrow H^{p+q}(X;F)$$ for abelian sheaves $F$ on a site $X$ and morphisms of sites $f : X \to ...
2
votes
0answers
53 views

Global sections of a tangent sheaf of a blown-up surface.

Let $V$ be a smooth projective algebraic surface and let $\pi\colon V'\to V$ be a blowup of a point $p\in V$. I would like to ask if the following is true: $h^{0}(V,\Theta_{V})=h^{0}(V', ...
2
votes
0answers
37 views

Subring of all elements represented by quotients of function field.

Suppose $K_C$ is the function field of a curve $C$ and $p \in C$. Let $\mathcal{O}_k \subset K_C$ be the subring of all elements represented by quotients $G/H$ where $G, H \in \mathbb{C}[x, y, z]$ are ...
2
votes
1answer
33 views

Natural isomorphism between curve and its projective completion?

If $C \subset \mathbb{C}^2$ is an irreducible affine curve and $\hat{C} \subset \mathbb{P}_2$ is its projective completion, is there necessarily a natural isomorphism of function fields $K_C \cong ...
3
votes
1answer
48 views

Subsheaf generated by one section is coherent

I'm working on exercise II.5.15 in Hartshorne's book. I need to prove the following bit. Let $ X $ be a noetherian scheme. Let $\mathscr {F }$ be a quasi coherent sheaf on $ X$. Then the subsheaf ...
3
votes
1answer
47 views

How can affine coordinate rings be canonically identified as $k$-algebras?

Exercise 1.5 of Hartshorne asks us to show (in one direction) that any affine coordinate ring $k[x_1,\dots,x_n]/I(Y)$ is a finitely-generated $k$-algebra with no nilpotents. The second part is quite ...
0
votes
1answer
48 views

How many ellipsoids can be maximally inside a circle?

This discussion is related to this discussion here where I want to deduce the area difference between such two circles filled with ellipsoids. Actually, to understand this difference is the main ...
0
votes
0answers
32 views

Smooth morphism and completion of DVR

Let $R$ be a Henselian discrete valuation ring with algebraically closed residue field and $\hat{R}$ its $m$-completion, where $m$ is the maximal ideal. Is it true that the induced morphism ...
0
votes
0answers
32 views

DVR and its fraction field

Let $k$ be a complete discrete valuation field with algebraically closed residue field. We know that its maximal unramified extension $k^{\mathrm{unr}}$ need not be complete. But can the ring of ...
1
vote
0answers
54 views

Local ring at generic point

Let $X$ be a smooth projective variety, and $Y$ a subvariety of codimension one (both are irreducible). I want to show that the local ring $\mathcal{O}_{Y,X}$ at the subvariety $Y$ (which is nothing ...
1
vote
1answer
58 views

If $\mathcal O_P(C)$ is a DVR, then $P$ is non-singular

Let $C$ be an irreducible curve over $\mathbb A^2$ and $P\in C$. I would like to prove if $$\mathcal O_P(C)=\{f\in k(C)\mid f=a/b, b(P)\neq 0\}$$ is a DVR, then $P$ is non-singular, i.e., the ...
5
votes
1answer
54 views

Proof that the set of intersection points is finite

I am trying to understand the proof of the following theorem: Let $f,g\in K[x,y]$ without a common factor. Then $\#V(f,g)<\infty$. (Here $K$ is a field and ...
2
votes
0answers
42 views

Cohomologies of $\Omega^k_{\mathbb{P}^n}$ and $\bigwedge\nolimits^k T_{\mathbb{P}^n}$

How to compute cohomologies of the sheaves of differential $k$-forms $\Omega^k_{\mathbb{P}^n}$ and exterior power of the tangent sheaf $\bigwedge\nolimits^k T_{\mathbb{P}^n}$? I am interested in ...
0
votes
0answers
31 views

Circle Rolling on Ellipse

I've gotten interested in describing a circle rolling on an ellipse, specifically the curve traced out by a point on the circumference of the circle. I want a symbolic solution to the general case, ...
1
vote
1answer
58 views

Two definitions of coherent sheaf

There are at least two ways to define a (quasi)coherent sheaf on the scheme $(X,\mathcal{O}_X)$, namely the one given in Hartshorne's "Algebraic geometry" (I guess you know it: it is given in terms of ...
2
votes
2answers
89 views

What can be said about a regular quotient (by a principal prime ideal) of a polynomial ring?

Let $f \in \mathbb{C}[x_1,\ldots,x_n]$ be irreducible (so (f) is a prime ideal). Assume $S:=\mathbb{C}[x_1,\ldots,x_n]/(f)$ is regular, where, by definition, a noetherian ring is regular is all its ...
1
vote
1answer
52 views

Confusion about affine schemes

Let $\varphi:A \rightarrow B$ be a ring morphism. Consider the corresponding map of schemes, $f: Spec\ B \rightarrow Spec\ A$. This map $f$ is given by $\varphi ^{-1},$ since prime ideals pull back ...
2
votes
1answer
47 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 $\eta$ ...
2
votes
0answers
50 views

Calculating intersection with diagonal in $\mathbb{P}^2 \times \mathbb{P}^2$

The following is example 6.1.2 from Fulton, Intersection Theory. Denote projective coordinates on $\mathbb{P}^2$ by $[x,y,z]$, and on $Y=\mathbb{P}^2 \times \mathbb{P}^2$ by $([x,y,z],[u,v,w])$. ...
0
votes
0answers
65 views

What exactly is a conifold for mathematicians?

For physicists a conifold is a generalization of a manifold that has a singular point. For example the resolved conifold is the space given by the solutions of $$xy-zt = 0$$ where $(x,y,z,t)\in ...
3
votes
0answers
26 views

$D$ is divisor of both $d(x/z)$ and $y/z$. [closed]

Let $C \subset \mathbb{CP}^2$ be the cubic curve defined by$$y^2z = x(x-z)(x-\lambda z)$$with $\lambda \in \mathbb{C} - \{0,1\}$. Let $p = [0, 0, 1]$, $q = [1, 0, 1]$, $r = [\lambda, 0, 1]$, and $s = ...
0
votes
1answer
63 views

How to tell if (X,Y) coordinate is within a Circle

Lets say we have a circle on a MxN grid as shown below. How can we determine whether the coordinate X,Y falls within the circles coordinates under the assumptions? We know the diameter of the ...
7
votes
0answers
65 views

Relationship between twisting sheaves and divisor sheaves

I'm not really entirely sure how to think about Serre's twisting sheaves $\mathscr{O}(i)$ - on any $\text{Proj}$ construction, really, but let's stick to something like $\mathbb{P}_2$ for now for ...
1
vote
1answer
92 views

How to prove $\mathcal O_P(C)$ is a DVR for $P$ non-singular?

Let $C$ be an irreducible curve over $\mathbb A^2$ and $P\in C$ a non-singular point. I want to prove that $\mathcal O_P(C)=\{f\in k(C)\mid f=a/b, b(P)\neq 0\}$ is a DVR. I've already proved that ...
3
votes
0answers
50 views

$X$ and $Y$ are schemes of finite type over $S$, then $X \times_S Y$ is of finite type over $S$

If $X$ and $Y$ are schemes of finite type over $S$, then $X \times_S Y$ is of finite type over $S$. I was reading a solution but it's not complete. Attempt of proof Let $f:X\to S$ and $g:Y\to S$ be ...