Let L be a very ample line bundle on a smooth variety X. Let D be the set of sections of L whose zero loci are singular. Then D is a divisor in H^0(L), the space of sections of L. Can D contain a hyperplane of H^0(L)?

Embedding $X$ in $\mathbb{P}(H^0(L)^*)$, this means that every hyperplane through a certain point $Q$ is tangent to $X$. In other words, $X$ must be a hypersurface, and every hyperplane tangent to $X$ contains $Q$. Conics in characteristic two are an example, and, in fact, the only one in dimension 1 (E. Lluis, Bol. Soc. Mat. Mexicana (2) 7 (1962) 47–56; MR0147479 (26 #4995)). I don't know what is known in higher dimensions. 

