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Let $X=\mathbb{P}^2$, and let $(y_1,y_2,y_3)$ be homogeneous coordinates on $X$. Consider a map $\phi:\mathbb{P}^1\longrightarrow X$, given by $\phi(x_1,x_2)=(x_1^2,x_1x_2,x_2^2)$, where $(x_1,x_2)$ are coordinates on $\mathbb{P}^1$. Then the image is the conic in $\mathbb{P^2}$, given by $V(y_2^2-y_1y_3)$. And $i^*\mathcal{O}_X(1)=\mathcal{O}_{\mathbb{P}^1}(2)$, and therefore has degree 2.

Can we same something analogous for every curve $C$ in $\mathbb{P}^n$? That is is can we identify it as a the image of a morphism under $\mathbb{P}^1$, and hence define the degree of $i^*\mathcal{O}(1)$ as the degree of the pulled back line bundle in $\mathbb{P}^1$?

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  • $\begingroup$ What is your definition of degree? Some people may define degree as $\deg(i^\ast\mathcal{O}(1))$. $\endgroup$ – Alex Youcis Jul 21 '15 at 13:58
  • $\begingroup$ Yes I would like to know the $deg(\mathcal{O}_C(1))= deg(i^*\mathcal{O}_X(1))$ $\endgroup$ – gradstudent Jul 21 '15 at 14:04
  • $\begingroup$ How else can one define degree? Thanks in advance! $\endgroup$ – gradstudent Jul 21 '15 at 14:05
  • $\begingroup$ Look at Riemann-Roch? $\endgroup$ – Hoot Jul 21 '15 at 14:07
  • $\begingroup$ Thanks Hoot, I would like to know how the pull back affects the degree? How do I use the Riemann roch? I don't know the gebus, I don't know $h^O$ and $h^1$? $\endgroup$ – gradstudent Jul 21 '15 at 14:11

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