An alternating trilinear form on a vector space \(V\) over a field \(\mathbb F\) is a mapping \(T:V^3\to{\mathbb F}\) which is linear separately in each variable and satisfies \(T(v_1,v_2,v_3)=0\), whenever \(v_i=v_j\) for distinct indices \(i, j\).
The authors investigate the existence of alternating trilinear forms with some special properties regarding their zeros. By passing to quotient spaces, one can assume from the start that the form is nondegenerate, i.e., that for each nonzero \(a\in V\) there exist \(b,c\in V\) with \(T(a,b,c)\neq0\). However, \(T\) may also vanish on a projective line, that is, the functional \(T(a,b,\cdot)\) may vanish identically for two distinct points \([a],[b]\) in a projective space \({\mathbb P}V\). In case of finite-dimensional real vector spaces, it is shown that except when \(\dim V\in\{3,7\}\) every alternating trilinear form on \(V\) has this property. The exception at \(\dim V=7\) is related to the existence of eight-dimensional real division algebra of octonions. It is further shown that six-dimensional vector spaces over finite fields admit trilinear alternating forms such that through each point \([a]\in{\mathbb P}V\) there exists a single projective line \([a,b]\subseteq{\mathbb P}V\) with \(T(a,b,\cdot)\) vanishing identically.