Geometric Points in Tensor Triangular Geometry

In this paper, we study geometric points in tensor triangular geometry. In doing so, we construct a counter-example to Balmer’s Nerves of Steel conjecture using free constructions in higher Zariski geometry. We then go on to introduce and discuss constructible spectra in the context of tensor triangular geometry. For tensor triangulated categories satisfying a mild enhancement condition, we use these spectra to construct geometric incarnations of (homological or triangular) primes via maps to “pointlike” tensor triangulated categories.