I take an irreducible and reduced closed curve $C\subseteq \mathbb{P}^n$, defined over an algebraically closed field $k$ and define the arithmetic genus $p\_a(C)$ as the integer such that the Hilbert polynomial of $C$ (or of its ideal of polynomials vanishing on it) is
$$h\_C(X)=\deg(C) X+1-p\_a(C).$$
**I would then like to have a simple proof that $p\_a(C)\ge 0$.** Of course, one may say that it is because it is the dimension of a vector space obtain using cohomology, but I would like **a proof not involving cohomology**, as it is for a master work and in my course of algebraic geometry I (unfortunately you will say) did not define the cohomology. It would then take a lot of time for the student to understand the cohomology just to prove this.


In the course, we defined the Hilbert Polynomial of any ideal, computed local intersections, Bézout theorem in $\mathbb{P}^n$ and studied blow-ups of surfaces. We also proved that $p\_a(C)=g(C)$ when $C$ is smooth, where $g(C)$ is given by Riemann-Roch (the smallest integer such that $\ell(D)\ge \deg(D)+1-g$ for each divisor $D$ on $C$). I would like the student to use the fact that $p\_a(C)\ge 0$ to bound the type of singularities of a plane curve and to show that one can have a resolution by blowing-up the singular points and repeating this process finitely many times.


Thanks for your help.



Simple proof that the arithmetic genus is non-negative
Mumford (*Complex Projective Varieties*, section 7) has the following, reasonably simple proof.
Let $d$ be the degree of $\mathrm{C}$, $m$ big enough such that $h\_{\mathrm{C}}(m)=\mathrm{dim}\_{\mathbf{C}} (\mathbf{C}[\mathrm{T}\_0,\ldots,\mathrm{T}\_n]/\mathrm{I}(\mathrm{C}))\_m$ and $md/2>p\_a$. Embed $\mathrm{C}$ into $\mathbf{P}^N$ by the degree $m$ Veronese embedding; let $\mathrm{L}\subset\mathbf{P}^N$ be a linear space containing $\nu\_m(\mathrm{C})$ such that $\nu\_m(\mathrm{C})$ is nondegenerate in $\mathrm{L}$. Then $\dim(\mathrm{L})=h\_\mathrm{C}(m)=md+1-p\_a$, in other words $\mathrm{L}\simeq\mathbf{P}^{md-p\_a}$, and the degree of $\nu\_m(\mathrm{C})$ is $$md=\mathrm{deg}(\nu\_m(\mathrm{C}))\geqslant\mathrm{codim}(\nu\_m(\mathrm{C}))+1\geqslant md-p\_a$$
since $\nu\_m(\mathrm{C})$ is nondegenerate in $\mathrm{L}$.


