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Let $X\subset\mathbb{P}^2$ be a complex manifold defined by a homogeneous polynomial of degree $d>3$. Let $$\phi:\mathbb{P}^1\rightarrow X$$ be a holomorphic map. Show that $\phi$ is constant.

Let $\mathcal{O}_{\mathbb{P}^n}(-1)$ be the Tautological line bundle on $\mathbb{P}^n$. We know the following facts:

  • The Canonical line bundle of X is the restriction of $\mathcal{O}_{\mathbb{P}^2}(d-2-1)=\mathcal{O}_{\mathbb{P}^2}(d-3)$ to $X$.
  • The Canonical line bundle of $\mathbb{P}^1$ is $\mathcal{O}_{\mathbb{P}^1}(-2)$.

From this we can conclude that $X$ and $\mathbb{P}^1$ are certainly not isomorphic.

Now the problem is solved straightforwardly by means of the Riemann-Hurwitz formula:
if $\phi$ is non constant then the genus of $X$ is less or equal the genus of $\mathbb{P}^1$, which is zero. Thus $g_X=0$, and we know this implies $X$ to be isomorphic to $\mathbb{P}^1$. This is a contradiction.

My goal is instead to solve the problem without using the theory of Riemann Surfaces but by means of other methods, e.g. the properties of holomorphic mappings.

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