Let \(\ts{X_t}\) be a family of hypersurfaces of the same degree in \(\PP_k^n\). Show that for each \(i\) the function \(h^i(X_t, \mco_{X_t})\) is a constant function of \(t\).
Solution
Let \(d\) be the common degree of the hypersurfaces.
1. For every parameter value \(t\), the hypersurface \(X_t\subseteq\PP_k^n\)fits into an exact sequence\[ 0 \longrightarrow \mco_{\PP^n}(-d) \xrightarrow{\cdot F_t} \mco_{\PP^n} \longrightarrow \mco_{X_t} \longrightarrow0, \]where \(F_t\) is a homogeneous equation of degree \(d\) for \(X_t\).
A hypersurface of degree \(d\) is the effective Cartier divisor cut out by one nonzero homogeneous form \(F_t\) of degree \(d\). Its ideal sheaf is therefore \[ \mci_{X_t}\cong\mco_{\PP^n}(-d), \] with inclusion into \(\mco_{\PP^n}\) given by multiplication by \(F_t\). Taking the quotient gives the displayed sequence.
2. The cohomology groups of the two line bundles in step 1 depend onlyon \(n\) and \(d\):\[ H^q(\PP^n,\mco)= \begin{cases} k,&q=0,\\ 0,&q>0, \end{cases} \]and\[ H^q(\PP^n,\mco(-d))=0 \qquad (0<q<n). \]Moreover\[ h^n(\PP^n,\mco(-d)) = \begin{cases} \binom{d-1}{n},&d\ge n+1,\\ 0,&d\le n. \end{cases} \]
These are the standard cohomology formulas for twisting sheaves on projective space, recorded in the cohomology of $\mco_{\PP^n}(d)$.
3. Assume \(n\ge2\). Then for every \(t\),\[ h^0(X_t,\mco_{X_t})=1, \]\[ h^q(X_t,\mco_{X_t})=0 \qquad (0<q<n-1), \]and\[ h^{n-1}(X_t,\mco_{X_t}) = \begin{cases} \binom{d-1}{n},&d\ge n+1,\\ 0,&d\le n, \end{cases} \]and\[ H^q(X_t,\mco_{X_t})=0 \qquad (q\ge n). \]
1. Since\[ H^0(\PP^n,\mco(-d))=0 \]and, for \(n\ge2\),\[ H^1(\PP^n,\mco(-d))=0, \]the beginning of the long exact sequence gives\[ H^0(X_t,\mco_{X_t})\cong k. \]For \(0<q<n-1\), both neighbouring projective-space groups vanish bystep 2, so\[ H^q(X_t,\mco_{X_t})=0. \]Finally the relevant tail is\[ 0 \longrightarrow H^{n-1}(X_t,\mco_{X_t}) \longrightarrow H^n(\PP^n,\mco(-d)) \longrightarrow H^n(\PP^n,\mco) =0. \]Thus\[ H^{n-1}(X_t,\mco_{X_t}) \cong H^n(\PP^n,\mco(-d)), \]whose dimension is the value in step
2. The same long exact sequencegives \(H^q(X_t,\mco_{X_t})=0\) for \(q\ge n\). None of these dimensionsdepends on \(t\).
4. If \(n=1\), then for every \(t\),\[ h^0(X_t,\mco_{X_t})=d \]and\[ H^q(X_t,\mco_{X_t})=0 \qquad (q>0). \]
The long exact sequence of step 1 becomes \[ 0 \longrightarrow k \longrightarrow H^0(X_t,\mco_{X_t}) \longrightarrow H^1(\PP^1,\mco(-d)) \longrightarrow0, \] because \(H^0(\PP^1,\mco(-d))=H^1(\PP^1,\mco)=0\). By the projective-line case, \[ h^1(\PP^1,\mco(-d))=d-1 \] for \(d\ge1\). Hence \[ h^0(X_t,\mco_{X_t})=1+(d-1)=d. \] A hypersurface in \(\PP^1\) is zero-dimensional, so its higher coherent cohomology vanishes.
5. For every \(i\), the number\[ h^i(X_t,\mco_{X_t}) \]is independent of \(t\).
For \(n\ge2\), step 3 gives every cohomology dimension explicitly in terms of \(n\) and \(d\). For \(n=1\), step 4 does the same. If \(n=0\), a positive-degree hypersurface in \(\PP^0\) is empty, so every cohomology group is zero. Thus fixing \(i\), the function of the parameter \(t\) is constant.
6. Q.E.D.
Step 5 is the required conclusion.