Let $X$ be a scheme over $\mathbb C$ carrying a *symmetric* perfect obstruction theory $\phi:E\to L_X$, in the sense of Behrend-Fantechi (here $L_X$ is the truncated cotangent complex, in degrees $[-1,0]$). Symmetry implies that the obstruction sheaf is $\textrm{ob}=\Omega_X$. I would like to understand the notion of *obstruction cone* attached to the obstruction theory.

It should be the image of the intrinsic normal cone $\mathfrak C_X$ under the chain of closed immersions $$\mathfrak C_X\subset \mathfrak N_X\subset \mathcal E=h^1/h^0(E^\vee).$$ The second inclusion is $h^1/h^0(\phi^\vee)$, and is closed by definition of perfect obstruction theory, and the first one is always there (the intrinsic normal sheaf is the abelian hull of the intrinsic normal cone). So we have the obstruction cone $$\mathfrak C\subset \mathcal E.$$

Now, in this paper by Behrend I found another notion of obstruction cone, and I would like to understand why it is the same as the one I just described.

Embedding $X$ in a smooth scheme $M$ (let us assume we can do this), we get a surjection $$p:\Omega_M|_X\twoheadrightarrow \textrm{ob}=\Omega_X.$$

Behrend talks about the cone of curvilinear obstructions $i:\textrm{cv}\hookrightarrow\textrm{ob}$ and says that the obstruction cone is the fiber product of the arrows $p$ and $i$.

Now, there are at least two things I do not understand: first of all, what exactly is the cone of curvilinear obstructions? Apparently it can be identified with the coarse moduli *sheaf* of $\mathfrak C_X$, but I have no idea how the latter is defined. Secondly, why are the two descriptions of the *obstruction cone* equivalent? (Probably the second one is the local version of the first, but still I cannot see how they relate to one another).

Thanks!