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For a general curve C of genus $g\geq 10$, we show that the Brill- Noetherlocus $B^4(2,K_C)$ contains irreducible sub-varieties $B_3\supset B_4 \supset\cdots \supset B_n$, where $B_n$ is of dimension $3g-10-n$ and $B_3$ is anirreducible component of the expected dimension $3g-13$.

An stratification of $B^4(2,K_C)$ over a general curve