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The intersection of two varieties is given by the sum of their ideals. So you're looking at the ideal generated by the union of the generators of $V_V$ and $V_S$. Immediately we see that both $x_1^2-x_0x_3$ and $x_0x_3-x_1x_2$ are generators of $I=I_{S \cap V}$. But this implies that $x_1^2-x_1x_2=x_1(x_1-x_2)$ is a generator of $I$. Thus $I$ is reducible. ...

3

There is a bijection between the set of lines and hyperplanes in any vector space, but it is not canonical (it depends on a choice). Namely, choose a non-degenerate bilinear form (this you can do for instance by choosing a basis and taking the standard non-degenerate bilinear form with respect to it). Then the orthogonal complement of a line is a ...

3

My solution, for now: We take the line $l$ through $BT\cap AC,AT\cap BC$, $D=l\cap AB$, then $DC$ is tangent to the ellipse at $C$; Let $E=AT\cap DC$ and $\Gamma$ be the circle tangent to $AT,BT$ at $A,B$; Let $F=CT\cap\Gamma$ (one of the two intersections); Let $G$ be the intersection between $AT$ and the tangent to $\Gamma$ at $F$; Let $H$ be one ...

2

This is false: let $C = \mathbb{P}^1_{\mathbb{C}}$ and let $L = \mathcal{O}(-1)$ and $L' = \mathcal{O}(1)$. Then, $$2 = \underbrace{h^0(C,L)}_{=0} + \underbrace{h^0(C,L')}_{=2} \neq h^0(C, L \otimes L') = 1.$$ It might be more reasonable to expect the dimensions to multiply i.e. have $$h^0(C,L) \cdot h^0(C,L') = h^0(C,L\otimes L'),$$ because there is a ...

1

There is a bijection between the hyperplanes and the projective points. But defining in terms of hyperplanes is strange (as mentioned in another answer, it is the dual; where did you find this definition, by the way?). Projective geometry is sometimes defined, not in terms of the one-dimenional spaces, but in terms of ALL the subspaces of $V$. This gives ...

1

The line through $T$ and the midpoint of $\overline{AB}$ passes through the center of the ellipse. (In fact, it's one of the axes, which will be important later.) Moreover, if the tangents at $B$ and $C$ meet at, say, $U$, then the line through $U$ and the midpoint of $\overline{BC}$ also passes through the center. With two lines pinpointing its location, we ...

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This question has been asked and answered on MathOverflow. I have replicated the accepted answer by Dmitri below. What you want almost works, but with a little twist. The group $SO(3)$ is acting on $F_2$, so that there are two orbits that are $\mathbb CP^1$ and all other orbits are $SO(3)$. More precisely, $F_2$ can be seen as a compactification of ...

1

$\phi$ is degree one if and only if it is an isomorphism from an open subset of $X$ to an open subset of $Y$ (i.e. it is birational). So it is generically one-to-one. But birational maps can be far from injective. As counterexamples, consider blowups of varieties along closed subvarieties. These are always birational but never injective. What you can say ...

1

Here is a construction that is not "tricky" or perhaps elegant but is straightforward. In your desired formula, let's replace the segment length $RS$ with the equivalent $(PS-PR)$, so we have only one segment length that is dependent on point $R$. Then we get $$\frac{AC \cdot BD} {AB \cdot CD} = \frac{PR \cdot QS} {PQ \cdot (PS-PR)}$$ The only unknown ...

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Sorry to resurrect my old question, but for the sake of completeness, I will provide a complete solution. Consider the tangent space at the origin of our divisor $D$. We have$$k[D] = k[a, b, c, d]/(a, b, ac - bd) \cong k[c, d].$$This Zariski cotangent space has dimension $2$ here (generated by $c$, $d$). Now, let us consider a tangent space of $X \cap ... 1 For 2), let$\pi:X=\mathbb{P}(E)\to S$be the natural map. We have a canonical surjection$\pi^*E\to\mathcal{O}_X(1)$and thus a map by composition$\pi^*E'(-1)\to\mathcal{O}_X$. The image is the ideal sheaf defining$Y=\mathbb{P}(E")$. 1 Mariano in the comments makes a very good suggestion to look at Veblen and Young. But the following axiomatic description is the heart of it: A projective space$\mathbb{P}\$ is a set of objects of two types. An object of the first type is called a point; one of the second type is called a line. A point and a line may or may not be incident, and the ...

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