Tagged Questions

Projective geometry is closely related to perspective geometry. These types of geometry originated with artists around the 14th century.

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Prerequisite of Projective Geometry for Algebraic Geometry

I studied Euclidean Geometry in high-school, and I have not studied anything relates to geometry since I started studying in university. I am now intending to study Algebraic Geometry, however, I ...
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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 ...
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Weighted blow-ups

I would like to understand what's a weighted blow-up in a very simple case: $\mathbb{C}^2$ blown-up in the origin with weights $(a,b)$. In found some notes online saying that this is the surface $X$ ...
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Harris, Exercise 10.28 (weighted projective spaces)

So I recently started teaching myself about weighted projective spaces from Harris' Algebraic geometry. It was going well until I came across this exercise, which has me stumped: "Show that any ...
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Is this method of finding a “dual curve” correct?

I have a very limited exposure to projective geometry, but I'm having fun exploring the concept of duality. In particular I'd like to know if this naive method of finding a "dual" curve to a given ...
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Find the intersection of two lines entirely outside the given sheet of paper by straightedge alone

This is a problem from Courant:"Two straight lines entirely outside the given sheet of paper are each given by two pairs of straight lines intersecting at points of the lines outside the paper. ...
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Literature request: Method for constructing projective manifolds

Currently (background: I'm preparing to write a thesis in mathematical physics) I'm quite often encountering a certain method for constructing projective manifolds, where the space is specified by ...
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Is the union of two projective curves in the projective plane a projective curve?

As the title suggests, is the union of two projective curves in the projective plane a projective curve? Any help would be appreciated, thanks.
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A surprising locus of points

I was playing with the setup for Desargues Theorem in Geogebra today and got a very odd-looking (to me) result. I imagine I could grind through this analytically and get an ugly-looking parametric ...
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Describing $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(m))$ as gluing of affine charts

How can we describe $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(m))$ as a gluing of affine charts? I'm having trouble with this problem, perhaps because I don't understand ...
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Properties of divisors when moving from char 0 to char p.

Consider a smooth projective variety $X$ over $\mathbb{C}$ such that $X$ has models over $\mathbb{Z}[1/N]$ and $X_p=X_{\mathbb{Z}[1/N]}\times \text{Spec}(\mathbb{F}_p)$ is also a smooth projective ...
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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 ...
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Tricks to find $\dim H^0(O(D)(n))$

Let $X$ be a curve of genus $g\neq 0$ and $D$ a divisor on $X$. If $n \in \mathbb{Z}$ and $O(D)(n)=O(D)\otimes O(n)$ is the twist of $O(D)$ by $n$, there is a manageable proceedings to find the ...
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A generalization of Zeeman-Gossard perspector theorem

I found a conjectures of generalization of the Zeeman Gossard theorem a year ago, but I no found a solution for the conjecture. I'm an electrical engineer, I am not a mathematician. I don't know how ...
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Invariant points and lines under homography

Given a matrix representation of an homography in a real projective space $P(\mathbb{R^3})$, what is the general procedure to calcule the invariant subspaces? A brief description would be enough.
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Irreducible closed subsets of projective varieties

I want to prove the following lemma: Let $X \subset \mathbb{P}^n$ be a projective variety. Let $W \subset X$ be a closed irreducible set. Then $W$ is also a projective variety. My idea is as ...
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Do we have homogeneous coordinates for probabilities?

As a roboticist, implementing visual odometry on a robot, homogeneous coordinates are convenient for projections of a non-moving object on an image sensor at $t$ and $t+1$ to estimate its position, ...
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Complex projective space CP3 (Twistor space), as bundle space with base CP1, and fiber 4-D Minkowski space-time?

Twistor space, as complex projective space $CP3$, is related to Minkowski 4-D space time (metric $1, -1,-1,-1)$, by the incidence relation. Let $Z = (v_a, u^{\dot{a}})$ a point in $CP3$, where $v_a$ ...
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There is some intuitive idea of Pascal's 's theorem in Projective Geometry?

In projective geometry, Pascal's theorem (formulated by Blaise Pascal when he was 16 years old) determines that a hexagon inscribed in a conic, the lines that contain the opposite sides intersect in ...
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Intersection of a cone $x^2+y^2-z^2$ and a generic plane in $\mathbb{RP}^3$

If we take the zero locus of $x^2+y^2-z^2$ to be our cone, I'd like to know how to go about finding the intersection of the cone and a generic plane $Ax+By+Cz+Dw=0$. The result will be a conic, but ...
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Smooth curve in a linear system- what does that mean?

Let $X$ be a projective variety. Let $L$ be an ample line bundle on $X$. I noticed the notation $C\in |L|_s$, and they say that it denotes a smooth curve $C\in |L|$. There are two questions: a) if $X$...
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counting argument for unitals in project planes

Could someone explain the counting argument for how a unital can be embedded in a projective plane, could somone elaborate on the argument in wikipedia?
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Inverting an isometric projection?

I'm trying to invert a function that takes points on a 2-d plane to an isometric projection of that plane. This function is encoded as follows (as part of the Isomer library): ...