  We approach multipartite entanglement classification in the symmetric
subspace in terms of algebraic geometry, its natural language. We show that the
class of symmetric separable states has the structure of a Veronese variety and
that its $k$-secant varieties are SLOCC invariants. Thus SLOCC classes gather
naturally into families. This classification presents useful properties such as
a linear growth of the number of families with the number of particles, and
nesting, i.e. upward consistency of the classification. We attach physical
meaning to this classification through the required interaction length of
parent Hamiltonians. We show that the states $W_N$ and GHZ$_N$ are in the same
secant family and that, effectively, the former can be obtained in a limit from
the latter. This limit is understood in terms of tangents, leading to a
refinement of the previous families. We compute explicitly the classification
of symmetric states with $N\leq 4$ qubits in terms of both secant families and
its refinement using tangents. This paves the way to further use of projective
varieties in algebraic geometry to solve open problems in entanglement theory.
