Real points on complex plane curves (Q1105000)

Let \(X\subset {\mathbb{C}}P^ 5\) be a complex plane projective curve of degree m and assume that all real points \(X_{{\mathbb{R}}}=X\cap {\mathbb{R}}P^ 2\) are ordinary points of X. If there exists a real line which intersects X with multiplicity k at some point and if X intersects \({\mathbb{R}}P^ 2\) transversally then there are at least 2k-m real points on X. Corollary: Every complex curve of even degree \(m\) which has a singularity of multiplicity at least \(m/2+1\) has a real point. The proof is topological and makes use of some nice properties of the function \(\rho ([x_ 0:x_ 1:x_ 2])=((\sum ^{2}_{i=0}x^ 2_ i)\cdot (\sum ^{2}_{i=0}\bar x^ 2_ i))/(\sum ^{2}_{i=0}x_ i\bar x_ i)^ 2\) defined on \({\mathbb{C}}P^ 2\).