Conjecture on curve components of the log-free variety for even-degree divisors

Let $L = \sum_{i = 1}^{N} \ell_i p_i$ be a divisor with degree $\ell = \deg L$ even, and let $V$ denote the log-free variety associated with the polynomial system for the equation under consideration. A curve-component conjecture. The variety $V$ consists of a finite number of curves and points. Moreover, $V$ always has a non-trivial curve component $V_0 \subset V$. This conjecture extends the preceding observation from the primitive case to arbitrary even-degree divisors. The precise structure of the curve components is not established in the given text.