# Bezout Theorem in $\mathbb P^3$

For a $\mathbb C P^2$ is known a result: if through the generic points $p_1,p_2,\dots p_n$ with multiplicities $m_1,m_2\dots, m_n$ correspondingly a degree $d$ curve passes then $d^2\geq m_1^2+m_2^2+\dots+m_n^2-min (m_i)$. (see Lemma 1 in "Curves in $\mathbb P^2$ and symplectic packings" Geng Xu) The idea of the proof is the following: move slightly a point $p_i$ and apply Bezout theorem.

Can we do the same in $\mathbb CP^3$ ?

For a generic set of points $p_1,p_2,\dots p_n$ with prescribed multiplicities $m_1,m_2\dots m_n$ we are looking for a minimal degree $d$ hypersurface through them. Suppose it is $d$. Then slightly move $p_i$ in different directions and find two surfaces of degree $d$ through new collections of points.
Then apply Bezout theorem of these three hypersurfaces and get $d^3\geq m_1^3+m_2^3\dots +m^2_n- min(m_i)^2$.

The only problem is that these surfaces may intesect by a curve and we can not apply Bezout Theorem. Can we repair this?

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Hi Nikita, I do not know you "known result" for $\mathbb{P}^2$, could you give a reference? Given like this it is false, take for example $d=4$ and $m_1=m_2=\dots=m_5=2$. If you add the condition of irreducibility, maybe this could be true. I would be happy to see a proof of the result. – Jérémy Blanc Sep 1 '13 at 20:52
Hi Jérémy, sure, a curve is irreducible. I added a reference in the text – Nikita Kalinin Sep 2 '13 at 18:21
Maybe you could take a look at the first chapter of Rob Lazarsfeld's book Positivity in Algebraic Geometry, where he talks about intersection theory of divisors. – rfauffar Sep 2 '13 at 22:34
@Nikita. Thanks for the reference. – Jérémy Blanc Sep 3 '13 at 7:10