Let $n\ge 1$ and $d\ge 1$, and let $f\in \mathbb C[x_0,\dots,x_{n+1}]_d$ be a homogeneous polynomial such that

\begin{align*} X:=\{[x_0:\dots:x_{n+1}]\in \mathbb P^{n+1}:f(x_0,\dots,x_{n+1})=0\} \end{align*}

is a smooth complex projective hypersurface of dimension $n$. Define the graded Jacobian ring

\begin{align*} R_f:=\mathbb C[x_0,\dots,x_{n+1}]\big/\left(\frac{\partial f}{\partial x_0},\dots,\frac{\partial f}{\partial x_{n+1}}\right), \end{align*}

and write $R_{f,m}$ for its degree-$m$ graded piece.

Let $B$ be a local deformation space of $X$ through smooth degree-$d$ hypersurfaces, modulo infinitesimal projective automorphisms, and let $T:=T_XB$ be its Zariski tangent space. Suppose the Kodaira-Spencer map identifies $T$ with a linear subspace $V\subset R_{f,d}$. Let $D$ be the classifying period domain whose points are polarized Hodge filtrations on the fixed primitive middle cohomology [vector space](/page/Vector%20Space) $H^n(X,\mathbb C)_{\mathrm{prim}}$ with the same Hodge numbers and polarization type as $X$. Let

\begin{align*} \mathcal P:B\to D \end{align*}

be the local period map for the polarized primitive middle cohomology $H^n(X,\mathbb C)_{\mathrm{prim}}$, with the Griffiths residue convention identifying $H^{n-p,p}(X)_{\mathrm{prim}}$ with $R_{f,(p+1)d-n-2}$ for each integer $p$ with $0\le p\le n$ and identifying the infinitesimal period map with Jacobian-ring multiplication by the Kodaira-Spencer class.

Assume that there exists an integer $q$ with $0\le q\le n-1$ such that the multiplication representation

\begin{align*} \mu_q:V\to \operatorname{Hom}_{\mathbb C}\left(R_{f,(q+1)d-n-2},R_{f,(q+2)d-n-2}\right) \end{align*}

defined by $\mu_q(a)(b)=ab$ is injective. Then the differential

\begin{align*} d\mathcal P_X:T\to T_{\mathcal P(X)}D \end{align*}

of the primitive period map is injective.

In particular, the criterion applies to smooth plane curves of degree $d\ge 4$ with $q=0$ and to smooth cubic threefolds with $q=1$, by the Griffiths residue description of primitive cohomology and the corresponding Jacobian-ring multiplication theorem.