Equations of ${\bar{M}}_{0,n}$

Following work of Keel and Tevelev, we give explicit polynomials in the Cox ring of ${\mathbb{P}}^1\times \cdots \times {\mathbb{P}}^{n-3}$ that, conjecturally, determine ${\bar{M}}_{0,n}$ as a subscheme. Using Macaulay2, we prove that these equations generate the ideal for $n=5,6,7,8$. For $n\le 6$ we give a cohomological proof that these polynomials realize ${\bar{M}}_{0,n}$ as a projective variety, embedded in ${\mathbb{P}}^{(n-2)!-1}$ by the complete log canonical linear system.