Let $X$ be an algebraic scheme and $\mathscr C$ a cone on $X\times\mathbf A^1$ and $C_t$ denote the restriction of $\mathscr C$ to $X\times\{t\}$ ($t=0,1$, or whatever). The claim in Fulton's Intersection Theory is that when $\mathscr C$ is flat over $\mathbf A^1$, then $$s(C_0)=s(C_1)\in A_\ast X$$ where the Segre class $s(C)$ of a cone $C$ on a scheme $X$ is defined as $$s(C)=q_\ast\left(\sum_{i\geq0}c_1(\mathscr O(1))^i\smallfrown [P(C\oplus 1)]\right)$$ where $q:P(C\oplus 1)\to X$ is the projection.
Since if $C_i$ are the irreducible components of a cone $\mathscr C$ on $X$, then $s(C)=\sum_i m_is(C_i)$ and $[P(C\oplus 1)]=\sum_i m_i[P(C_i\oplus1)]$, where $m_i$ are the geometric multiplicities, so I believe we can restrict our attention to the class $[V]$ of an irreducible component of $\mathscr C$. In general a variety is flat over a nonsingular curve iff its generic point maps to the generic point of the curve. But if this is not true, then it means that the image of $V$ under projection to $\mathbf A^1$ is a point, say $t$. But in this case, $i_t^\ast(V)=0\ne V=V_t$, where I take $V_t$ to be the fiber and I take $i_t^\ast$ to be the Gysin morphism $A_\ast(X)\to A_{\ast-1}(D)$ where $C_t\subset\mathscr C$ is the divisor on $\mathscr C$ with local equation $t$ (aka the fiber). Since if $V=V_t$ then $i_t^\ast(V)$ is computed by restricting $\mathscr O_{\mathscr C}(C_t)$ to $V$ and then taking a corresponding Cartier divisor on $V$. But $\mathscr O_{\mathscr C}(C_t)$ restricted to $C_t$ is the normal bundle which is trivial, so $i_t^\ast(V)=0$.
So I see why flatness is needed. My question is, wouldn't the statement $$s(i_0^\ast\mathscr C)=s(i_1^\ast\mathscr C)\in A_\ast X$$ hold without the assumption of flatness? When $\mathscr C$ is flat over $\mathbf A^1$ it restricts to the original statement.
