0
votes
0answers
38 views

Injective and continuous function that is an embedding

Consider $n,d\in \mathbb N$ and $N= {n+d\choose d}-1$, then the well known $d$-uple embedding: $$\rho_d: \mathbb P^n(\mathbb C)\longrightarrow\mathbb P^N(\mathbb C)$$ is a continuous (respect to ...
2
votes
1answer
42 views

Blow-up in Projective Space: Choosing the Appropriate Chart

Consider $x^2+y+\alpha=0$ (made-up example), where $(x,y)\in\mathbb{C}^2$ and $\alpha\in\mathbb C$ is a free parameter. The equation defines a family of curves. The curves have no common points in ...
1
vote
2answers
39 views

Enumerative projective geometry

I am wondering whether for any two lines $\mathfrak{L}, \mathfrak{L'}$ and any point $\mathfrak{P}$ in $\mathbf{P}^3$ there is a line having nonempty intersection with all of $\mathfrak{L}, ...
3
votes
1answer
117 views

Blow-ups in Projective Space

This is in regards to a question (no solutions or comments thus far :-() I asked earlier in regards to the blow-up of an elliptic curve: Question Let $f(x,y)=y^2-4x^3+ax+b$, where $(x,y)\in\mathbb ...
2
votes
1answer
179 views

Resolution of Singularities: Base Point

Consider the curve $y^2=4x^3-ax-b$, where $a$ is a fixed constant and $b$ is a free constant. For each value of $b$ we get a family of curves. Part 1: Show that the family of curves intersect at ...
0
votes
1answer
51 views

Projective variety minus hyperplane $=$ affine variety

Claim: Let $V \subset \mathbb{C}P^n$ be a non-singular projective algebraic variety of complex dimension $k$ and let $P \subset \mathbb{C}P^n$ be a hyperplane. Then $V \setminus (V \cap P)$ is a ...
1
vote
0answers
43 views

On existence of a tangent line passing through a given point

Question Suppose $k$ is an algebraically closed field of characteristic $0$, and $C\subseteq\mathbb P^2(k)$ is an irreducible projective plane curve of degree $n>1$, and $P$ is a point on $\mathbb ...
1
vote
0answers
69 views

Intuition behind using projective geometry for defining the addition on an elliptic curve

We already had the chord-and-tangent construction that can be used to define a way of "adding" points on an elliptic curve. Also this addition satisfies all the group laws. Still why one needs to ...
2
votes
1answer
27 views

covering of projective curve by affine parts

For $\mathbb{P}^n$ we can let $U_i = \{(x_1:\cdots:x_i:\cdots:x_{n+1}) : x_i \neq 0\}$. Then let $C \subset \mathbb{P}^n$ be a projective plane curve. We can decompose $C$ into a union of affine plane ...
1
vote
1answer
48 views

Dimension of irreducible projective algebraic set

Let $Y \subset P^n(\Bbb{C}$) is an irreducible projective algebraic set, then how to show that dim$Y$ is equal to the minimum $r \in \Bbb{ N }$ such that there exists a linear subspace $S_{n-r-1} ...
1
vote
1answer
49 views

multiplicity of intersection of projective curves

This example is taken from Silverman and Tate's Rational Points on Elliptic Curves. Suppose we have the line $C_1: x+y=2$ and the circle $C_2: x^2 + y^2 = 2$. We see that these intersect at the ...
4
votes
1answer
39 views

Closed subschemes of projective space

We work over the complex numbers. If $X$ is an integral closed subscheme of $\mathbb P^n$, then there exists a homogeneous ideal $I$ of $A=\mathbb C[x_0,\ldots,x_n]$ such that $X = V_+(I) =$ ...
2
votes
1answer
39 views

The set of all $k$-dimensional planes which intersects $X$ is closed in $G(k,n)$

Let $X$ be irreducible algebraic set of projective n space. I am trying to show that: The set of all $k$-dimensional planes which intersects $X$ is closed in $G(k,n)$, where $G(k,n)$ is the ...
0
votes
2answers
54 views

Topological space underlying this curve

I have to solve this exercise but I have really no clue even how to start with it: Identify the topological space underlying the cubic $Y^2Z=X^2(X-Z)$ in $\mathbb{PR}^2$. How does it fit with the ...
2
votes
1answer
43 views

Calculating the projective closure with more than one generator

I am given a variety $X = Z(f_1,f_2)$ in affine 3-space (in $x,y,z$), and I would like to compute its projective closure $Y = Z(g_1,\dots,g_n)$ in projective 3-space (in $x,y,z,w$). I have seen this ...
0
votes
0answers
54 views

Dimension of embeddings of Segre variety (product of projective spaces)

The Segre map gives an embedding of the Segre variety $\Sigma_{n,m}$ (i.e. of the categorical product of two projective spaces of dimension $n$ and $m$) into a projective space of dimension $nm+n+m$. ...
2
votes
1answer
61 views

arithmetic and geometric genus for a reducible plane curve

If $C$ is an irreducible plane curve we have the well known formula relating the airthmetic (obtained via the degree-genus formula) and the geometric genus $$\frac{(d-1)(d-2)}{2} - \sum ...
2
votes
0answers
29 views

Projection from a point to a plane - confused about terminology.

Edit: It seems rude to delete the question, but I have my answer now thanks to rghthndsd. I'm a bit unsure about the terminology in a homework question I'm doing, and I can't find any clear answers ...
1
vote
1answer
42 views

Tangent lines of conics

Let $k$ be algebraically closed. Let $P\in k[x,y,z]$ be a homogeneous quadratic polynomial. Let $C$ be the zero locus of $(P)$ in $\mathbb{P}^2$. Let $Q \in \mathbb{P}^2$. Is there a tangent line at ...
2
votes
1answer
29 views

Show that $C\in\mathbb{P}^2(k)$ is a rational curve

Let $k$ be an algebraically closed field of characteristic $p>0$. We consider the curve $$C = V(X^pZ^{p-1}-Y^{2p-1})\subset\mathbb{P}^2(k)$$ Show that $C$ is a rational cuve. We did ...
5
votes
1answer
90 views

Distinguished open sets in $\mathbb{P}^n$

I have given a homogenous polynomial $F\in\mathbb{C}[x_0,\ldots,x_n]$ of degree $d>0$ and consider the open set $U_F=\{p\in\mathbb{P}^n; F(p)\neq 0\}$. I'd like to prove that $U_F$ is isomorphic to ...
3
votes
1answer
76 views

Imagining the projective Space

I am trying to get used to work in the projective space. Therefore I wanted to know which tactics there are to imagine the projective space. $$\mathbb{P^n}(k):= (k^{n+1}\backslash \{0\})/k^{*}$$ I ...
1
vote
0answers
57 views

Quotient of a proj variety by an involution

Usually, if you have an affine variety defined by some equations and have an involution on it, it's quite easy to immediately see what the equations of the quotient of the variety by the involution ...
4
votes
1answer
92 views

some fun with holomorphic line bundles

These are probably trivial questions... (for the experts) I'd like to get convinced (perhaps an intuitive/geometric explanation will be more effective than a formal one) of the following facts: i. ...
3
votes
1answer
55 views

Is the universal hyperplane section the blowup of the baselocus?

I think I've heard this statement before but I'd like to make sure it's true. Let $X$ be a variety and $L$ a line bundle on it. Take $S < P\left(H^0(X,L)\right)$ to be a linear subspace of the ...
3
votes
4answers
371 views

Two circles intersect in two points

Take for example two circles $$\begin{cases}x^2+y^2=1\\x^2+y^2-x-y=0\end{cases}$$ These two circles intersect in two points namely $(0,1)$ and $(1,0)$. But by Bezout's theorem they must intersect four ...
1
vote
1answer
59 views

Prove that a curve in P^n of degree n not contained in a hyperplane is rational

The set up is as stated above. We have a projective curve $X$ of degree n embedded in $\mathbb{P^n}$, which is not contained in any hyperplane. We claim that it is therefore rational. The way I have ...
3
votes
1answer
135 views

How to visualize projective plane

I need to comprehend projective plane as a prerequisite for some other topic. But I can't understand what it really looks like. How can I make this plane natural to me? Moreover I want to understand ...
2
votes
1answer
56 views

Smooth surface that is a complete intersection

I have this definition of a projective complex algebraic surface that is a complete intersection. A surface $S\subset\mathbb{P}^{r+2}$ is said to be a complete intersection if it is a trasversal ...
0
votes
1answer
66 views

Flatness question

In reading on the stacks project I came across a result I don't quite follow: "Assume M is finitely presented and flat, i.e., (1) holds. We will prove that (7) holds. Pick any prime p and x1,…,xr∈M ...
3
votes
2answers
110 views

Proving algebraically that $\mathbb RP ^3\cong SO(3,\mathbb R)$

I am giving a simple introductory course on algebraic geometry and I plan to mention that $$\mathbb RP ^3\cong SO(3,\mathbb R).$$ I know a rather simple proof of this using the fact that $\mathbb ...
2
votes
0answers
214 views

Good textbook on geometries

I am looking for a good textbook that thoroughly covers euclidean, affine, projective and non-euclidean geometries. I will be starting graduate school in algebraic geometry next year and I would like ...
4
votes
1answer
114 views

Difference between the real projective plane and the complex projective plane

Well the title says it all. If we consider the $P^2(\Bbb R)$ and the $P^2(\Bbb C)$, and we compare them, my guess is that it will be like a round $\Bbb R^2$ versus a sphere. I don't have very good ...
1
vote
0answers
36 views

Blowup of $\mathbb{P}^3$ along the ideal $(w^3 + x^3 + y^3 + z^3, w^4 + \alpha wxyz)$ for fixed $\alpha \in k$

I want to compute the blowup of $\mathbb{P}^3$ along the ideal $(w^3 + x^3 + y^3 + z^3, w^4 + \alpha wxyz)$ for fixed $\alpha \in k$. I've been working with blowups for a couple weeks, but this seems ...
10
votes
3answers
691 views

What is a good book to study classical projective geometry for the reader familiar with algebraic geometry?

The more I study algebraic geometry, the more I realize how I should have studied projective geometry in depth before. Not that I don't understand projective space (on the contrary, I am well versed ...
0
votes
0answers
69 views

Prove that irreducible curve of bidegree $(1, n)$ is rational for all $n \in \mathbb{N}$

I'm meant to prove that irreducible curve of bidegree $(1, n)$ is rational for all $n \in \mathbb{N}$. I have a proof for this statement that uses the genus formula to show that such a curve must ...
2
votes
0answers
52 views

Finding the equations of a variety under a projection map

Suppose I have a projective variety $X$ in $\mathbb{P}^N$ ($N >2$, say) defined as the zero set of some homogeneous polynomials $f_1, \ldots, f_r$. Consider the projection map $[x_0: x_1 : \cdots ...
4
votes
0answers
96 views

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 ...
3
votes
1answer
85 views

Blowup of $\mathbb{P}^n$ at a point is irreducible

The blowup of $\mathbb{P}^n$ at a point is irreducible. This seems clear intuitively, but I'm not sure how to prove it. Thoughts?
4
votes
0answers
89 views

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 ...
3
votes
1answer
83 views

Homeomorphisms of the cuspidal cubic

Are the cuspidal cubics $V(y^2-x^3)\subset \mathbb{A}^2$ and $V(X^3-Y^2Z)\subset \mathbb{P}^2$ homeomorphic to $\mathbb{A}^1$ and $\mathbb{P}^1$? I think I can see the homeomorphism in my mind (just ...
1
vote
0answers
23 views

Fibers of $V(ax_0^2+bx_1x_2)$ in $\mathbb{P}^1\times \mathbb{P}^2$

What are the fibers of $Z=V(ax_0^2+bx_1x_2)$ for $(a:b)\in \mathbb{P}^1, (x_0:x_1,x_2)\in \mathbb{P}^2$? If we fix $(a:b)$ or $(x_0:x_1,x_2)$ and dehomogenize, we can see that $Z$ is a family of ...
3
votes
2answers
88 views

Visualizing a projective variety

What does the variety $V(x_0^2+x_1^2+x_2^2)\subset \mathbb{P}^2$ look like? It seems to me like a single point... In general, are there any good ways/tips/tricks to visualize projective varieties?
0
votes
1answer
47 views

Classifying the different types of fibers on $V(uX^2 + vYZ) \subset \mathbb{P}^1 \times \mathbb{P}^2$

I'm working on an exercise where I'm supposed to note the different types of fibers on $V(uX^2 + vYZ) \subset \mathbb{P}^1 \times \mathbb{P}^2$. The way I was thinking to do this was to consider the ...
2
votes
0answers
18 views

Getting used to projective coordinates, need help describing (2) objects geometrically [duplicate]

I'm trying to get an intuition for what things look like in projective coordinates. There are two curves that I have to work a problem with, but I'm not sure how to visualize them. They are $V(u^2 X ...
1
vote
1answer
116 views

Birational isomorphism $\mathbb{P}^n\times \mathbb{P}^m\to \mathbb{P}^{n+m}$

One can show that $\mathbb{P}^n\times \mathbb{P}^m$ is birational to $\mathbb{P}^{n+m}$ by making note of the Zariski topologies and the canonical isomorphism between affine spaces $\mathbb{A}^n\times ...
4
votes
1answer
77 views

Description of varieties in $\mathbb{P}^2\times \mathbb{P}^1$

If $[x:y]$ are coordinates of $\mathbb{P}^1$ and $[X:Y:Z]$ are coordinates of $\mathbb{P}^2$, what do the following varieties look like? $V(x^2X+y^2Y+xyZ)\subset \mathbb{P}^2\times \mathbb{P}^1$ ...
3
votes
1answer
56 views

Can the method of resolvents be used to give a proof of Bezout's Theorem?

Can the method of resolvents be used to give a proof of Bezout's Theorem? It seems to me like it should but I am unable to finish the proof. Here is what I have so far. Take two homogeneous ...
2
votes
0answers
54 views

Are there in $(\mathbb{C}[x,y,z]/(x^3+y^3+z^3))_{x}$ exactly $12$ lines?

Let $R=\mathbb{C}[x,y,z]/(x^3+y^3+z^3)$ be the coordinate ring of the affine variety defined by the equation $x^3+y^3+z^3=0$. We can consider the localization in the element $x$, denoted by $R_x$. I ...
1
vote
1answer
96 views

Determining a conic section from points and tangent lines

Hi everyone can you please help me with this question? Is there no shorter way to do this then my approach? Is this a correct way to do it? Determine the conic section in $\mathbb{R}P^2$ that is ...