8
votes
1answer
70 views

Zariski vs analytic cohomology of $\mathcal O_X^\times$

Let $X/\mathbf C$ be a smooth proper variety. Is it true that $H^1(X, \mathcal O_X^\times) = H^1(X^{an}, \mathcal O_{X^{an}}^\times)?$ GAGA doesn't apply, because $\mathcal O_X^\times$ is not coherent ...
2
votes
1answer
43 views

Algebraic Compact manifold originates from a proper scheme?

If $M$ is a compact complex manifold, which is the analytification of some scheme $X$ of finite type over $\operatorname{Spec}(\mathbb{C})$, then must $X$ be proper over ...
2
votes
0answers
20 views

finite morphism (algebraic) vs finite morphism (analytic)

Let $X$ and $Y$ be two algebraic varieties (reduced schemes of finite type) over $\mathbb{C}$. Let $f : X \to Y$ be a morphism of schemes. Let $X^{an}$, $Y^{an}$ and $f^{an}$ the corresponding ...
4
votes
2answers
86 views

Applications of Stein spaces in Algebraic Geometry

I want to know where are essential applications of the theory of Stein spaces in algebraic geometry. I heard Cartan's theorem A & B were used in Serre's GAGA, but are there any other applications? ...
4
votes
0answers
61 views

Soft: Why does the existence of a singularity cause problems for deRham cohmology?

I've heard that if a variety has a singularity then the deRham theory has "problems". What exactly are these? Im guessing there is some sort of issue with the defintion of a differential form, but ...
0
votes
0answers
34 views

Smooth compact subvariety from a germ of an embedding?

I am interested in the extent to which a germ of an embedding determines a subvariety given certain global hypotheses. My intuition is that the answer should be yes under more general conditions than ...
1
vote
0answers
41 views

Question on a statement about analytic variety irreducible at $0$.

I am trying to understand this statement, it is in "Principles of Algebraic Geometry" by Philip Griffiths and Joe Harris. In page 13, third point, they are trying to prove that an analytic variety ...
5
votes
1answer
76 views

Is a flat morphism of complex algebraic varieties open in the analytic topology?

Let $X,Y$ be algebraic varieties over $\mathbb{C}$. A morphism $f:X\to Y$ induces a morphism $f^{an}:X^{an}\to Y^{an}$ between the associated complex analytic spaces (actually, I am interested only in ...
-1
votes
1answer
102 views

What is the polynomial satisfying $(t,t^2,t^3)$ [closed]

What is the equation satisfying the parameterized equation $(t,t^2,t^3)$ where $t$ is a real constant?
1
vote
1answer
97 views

Rank of Jacobian at a singularity

Is the following proposition true? Proposition: Suppose $\mathbb{C}\{x_1,\ldots,x_{d_1}\}/(f_1,\ldots,f_{k_1}) \cong \mathbb{C}\{y_1,\ldots,y_{d_2}\}/(g_1,\ldots,g_{k_2})$ are isomorphic complex ...
11
votes
2answers
430 views

When do equations represent the same curve?

Suppose we have two sets of parametric equations $\mathbf c_1(u) = (x_1(u), y_1(u))$ and $\mathbf c_2(v) = (x_2(v), y_2(v))$ representing two 2D planar curves. When I say "2D planar curves" I mean ...
1
vote
2answers
730 views

Finding the locus of the midpoint of chord that subtends a right angle at $(\alpha,\beta)$

There is a circle $x^2+y^2=a^2$. On any line that cuts the circle in two distinct points(it is a secant), the points of intersection with circle are taken and at those two points I draw the tangents ...
0
votes
1answer
144 views

In an equation that looks like the standard form of an ellipse, what must the constant on the RHS equal for exactly one solution?

I am working on a homework question: What must be the value(s) of $c$ for the following equation to have exactly 1 solution? The equation is of the standard form of the equation for an ellipse, ...
2
votes
1answer
133 views

Metric tensor of complex numbers & Hamiltonian Mechanics

The Euclidean $\mathbb{R}^2$ geometric space can be mapped onto $\mathbb{C}$. In other words I see it like this $$\vec{v} = x\vec{x}+y\vec{y} = x\vec{1}+y\vec{i}= \begin{bmatrix}x \\y\end{bmatrix} ...
2
votes
2answers
390 views

Is the area of intersection of convex polygons always convex?

I am interested specifically in the intersection of triangles but I think this is true of all convex polygons am I correct? Also is the largest possible inscribed triangle of a convex polygon always ...
5
votes
0answers
303 views

Proper mapping theorem

My professor mentioned a proper mapping theorem after the name of Remmert which says: Let $X$ and $Y$ be complex manifolds, $f:X \to Y$ be a proper holomorphic map, and $V \subset X$ be a complex ...
0
votes
1answer
89 views

Lines which intersect the postive half axis of x

We have to find out which lines intersect the postive half axis of x. According to this formula we can determine if the angle between two points(A[x1,y1] and B[x2,y2]) of the line (angle A0B where 0 ...
2
votes
0answers
75 views

Algebraic Geometry studied via Filters

Is there any research relating varieties with filters instead of radical ideals? For example, Suppose we have a variety V in C^n, now consider the fixed filter consisting of all algebraic sets ...
1
vote
1answer
120 views

Degree of Hessian surface invariant under linear transformations?

Given a surface $V(f) \subset \mathbb{P}^n$ for a homogeneous polynomial $f$ of degree $d$ on $\mathbb{P}^n$ and a linear transformation $g \in SL(n+1)$. Is the degree of the Hessian $H_f = V(\det ...
1
vote
1answer
93 views

Is the study of algebraic curve is techniquely equal to the advanced division of analytic geometry, if not, what is the difference?

Is the study of algebraic curve is techniquely equal to the advanced division of analytic geometry, if not, what is the difference? And what is other branch of advanced analytic geometry called? in ...
2
votes
0answers
42 views

degeneracy loci of dimension $2$

Let $X$ be a smooth complex projective variety of dimension $n \ge 4$ and let $F$ and $E$ be two (holomorphic) vector bundles of rank $f$ and $e$ over $X$. Given a morphism $\varphi: F \to E$ of ...
0
votes
1answer
343 views

algebraic way to compute intersection of disks

Is there a pure algebraic way to calculate intersection of two disks (extended to spheres, ellipses)?
1
vote
1answer
182 views

Cylinder silhouette

I have a parametric representation of a cylindrical shape (well, it's like a cone, but its spike is trimmed). I would like to have an analytic expression for its silhouette lines in terms of the ...
11
votes
1answer
187 views

algebraic versus analytic line bundles

If one has a quasiprojective complex variety X, there is a natural map from the algebraic Picard group to the analytic Picard group. Is this map either injective or surjective? I assume the latter ...
4
votes
3answers
462 views

The elementary coordinate geometry of polynomials? Of rational expressions? Of radicals?

With a few colleagues, we're trying to design an (intermediate) algebra course (US terminology) where we stress the interplay between algebra and geometry. The algebraic topics we would like to cover ...