Let \(Y\) be a nonsingular curve of degree \(d\) in \(\PP_k^n\), over an algebraically closed field \(k\). Show that \[ 0 \leq p_a(Y) \leq \frac{1}{2}(d-1)(d-2) .\]
Hint: compare \(Y\) to a suitable projection of \(Y\) into \(\PP^2\), as in (9.8.3) and (9.8.4).
Solution
Put \[ g=p_a(Y). \]
1. One has \(g\ge0\).
Because \(Y\) is a nonsingular projective integral curve over the algebraically closed field \(k\), \[ p_a(Y)=h^1(Y,\OO_Y) \] by the equality of the genera for a nonsingular projective curve. Therefore \[ \boxed{p_a(Y)\ge0}. \]
2. There is a linear projection\[ \pi:Y\longrightarrow C\subseteq\PP_k^2 \]which is birational onto an integral plane curve \(C\) of degree \(d\).
If \(n=2\), take \(C=Y\) and \(\pi=\id_Y\).
Assume \(n\ge3\). Choose a general linear centre \[ \Lambda\cong\PP^{n-3}\subseteq\PP^n \] disjoint from \(Y\). Successive general projections reduce the ambient dimension to \(3\) without identifying the generic point of \(Y\), and a general projection from \(\PP^3\) to \(\PP^2\) is birational on a curve. This is the generic-projection construction recorded in T-CRVEMBP3. Hence the restriction of the projection from \(\Lambda\) is a birational morphism onto an integral plane curve \(C\).
A general line \(L\subseteq\PP^2\) pulls back to a hyperplane of \(\PP^n\) containing \(\Lambda\). Since \(\pi\) has degree one on function fields, a general such hyperplane meets \(Y\) in exactly the points lying over \(C\cap L\), with the same total intersection multiplicity. Thus \[ \deg C=\deg Y=d. \]
3. The projection construction of III.9.8.3 gives a flat family\[ \mathcal Y\longrightarrow\AA^1 \]whose fibre over \(1\) is \(Y\) and whose fibre \(Y_0\) over \(0\) has support \(C\).
Choose homogeneous coordinates so that the projection of step 2 forgets the last \(n-2\) coordinates. For \(t\ne0\), scale those omitted coordinates by \(t\). The resulting projective automorphisms carry \(Y\) through an isotrivial family over \(\GG_m\). Taking its scheme-theoretic closure over \(\AA^1\) gives the flat family of Hartshorne III.9.8.3; the calculation in III.9.8.4 shows that its special fibre is a scheme supported on the projected curve \(C\).
Flatness preserves the Hilbert polynomial [@Har10a, Theorem III.9.9], hence \[ P_{Y_0}=P_Y. \] In particular \[ \deg Y_0=d \qquad\text{and}\qquad p_a(Y_0)=p_a(Y)=g. \]
4. The reduction of \(Y_0\) is \(C\), and the nilradical\[ \mathcal N=\ker(\OO_{Y_0}\longrightarrow\OO_C) \]has zero-dimensional support.
By step 3, the support of \(Y_0\) is \(C\), so \[ (Y_0)_{\mathrm{red}}=C. \] The exact sequence \[ 0\longrightarrow\mathcal N \longrightarrow\OO_{Y_0} \longrightarrow\OO_C \longrightarrow0 \] gives \[ P_{Y_0}=P_C+P_{\mathcal N}. \] Steps <1>2–<1>3 give \[ \deg Y_0=d=\deg C. \] For a one-dimensional projective scheme, the leading coefficient of the Hilbert polynomial is its degree. Hence the linear terms of \(P_{Y_0}\) and \(P_C\) cancel, so \(P_{\mathcal N}\) is constant. Therefore \(\mathcal N\) has zero-dimensional support. Since it is coherent on the projective curve \(C\), it has finite length.
5. The special fibre has arithmetic genus at most that of its reduced plane support:\[ p_a(Y_0)\le p_a(C). \]
There is an exact sequence \[ 0\longrightarrow\mathcal N \longrightarrow\OO_{Y_0} \longrightarrow\OO_C \longrightarrow0. \] Since \(\mathcal N\) is supported at finitely many closed points, \[ H^i(C,\mathcal N)=0\quad(i>0), \qquad \chi(\mathcal N)=h^0(C,\mathcal N)=\operatorname{length}(\mathcal N)\ge0. \] Additivity of Euler characteristic gives \[ \chi(\OO_{Y_0}) =\chi(\OO_C)+\operatorname{length}(\mathcal N). \] For a projective curve \(Z\), \[ p_a(Z)=1-\chi(\OO_Z). \] Consequently \[ p_a(Y_0) =p_a(C)-\operatorname{length}(\mathcal N) \le p_a(C). \]
6. The plane curve \(C\) has\[ p_a(C)=\frac12(d-1)(d-2). \]
By step 2, \(C\subseteq\PP^2\) is a plane curve of degree \(d\). The plane-curve Hilbert-polynomial calculation of Hartshorne I.7.2(b) gives \[ \boxed{p_a(C)=\binom{d-1}{2}=\frac12(d-1)(d-2)}. \]
7. The required upper bound follows.
By steps 3 and 5, \[ p_a(Y)=p_a(Y_0)\le p_a(C). \] Applying step 6 yields \[ \boxed{p_a(Y)\le\frac12(d-1)(d-2)}. \] Together with step 1 this proves \[ 0\le p_a(Y)\le\frac12(d-1)(d-2). \]
8. Q.E.D.
Step 1 proves the lower bound. Steps <1>2–<1>6 compare \(Y\) through the flat projection degeneration with a degree-\(d\) plane curve, and step 7 gives the upper bound.