Mori Dream Spaces as fine moduli of quiver representations

We construct Mori Dream Spaces as fine moduli spaces of representations of bound quivers, thereby extending results of Craw--Smith \cite{CrawSmith} beyond the toric case. Any collection of effective line bundles $\mathscr{L}=(\mathscr{O}_X, L_1,..., L_r)$ on a Mori Dream Space $X$ defines a bound quiver of sections and a map from $X$ to a toric quiver variety $|\mathscr{L}|$ called the multigraded linear series. We provide necessary and sufficient conditions for this map to be a closed immersion and, under additional assumptions on $\mathscr{L}$, the image realises $X$ as the fine moduli space of $\vartheta$-stable representations of the bound quiver. As an application, we show how to reconstruct del Pezzo surfaces from a full, strongly exceptional collection of line bundles.Comment: 25 pages, 2 figures; v2 section 3 simplified, typos corrected; v3 final versio

The Phenomenology of REM-sleep Dreaming: The Contributions of Personal and Perspectival Ownership, Subjective Temporality and Episodic Memory

Although the dream narrative, of (bio)logical necessity, originates with the dreamer, s/he typically does not know this. For the dreamer, the dream world is the real world. In this article I argue that this nightly misattribution is best explained in terms of the concept of mental ownership (e.g., Albahari, 2006; Klein, 2015a; Lane, 2012). Specifically, the exogenous nature of the dream narrative is the result of an individual assuming perspectival, but not personal, ownership of content s/he authored (i.e., “The content in my head is not mine. Therefore it must be peripherally perceived”). Situating explanation within a theoretical space designed to address questions pertaining to the experienced origins of conscious content has a number of salutatory consequences. For example, it promotes predictive fecundity by bringing to light empirical generalizations whose presence otherwise might have gone unnoticed (e.g., the severely limited role of mental time travel within the dream narrative)

Global Okounkov bodies for Bott-Samelson varieties

We use the theory of Mori dream spaces to prove that the global Okounkov body of a Bott-Samelson variety with respect to a natural flag of subvarieties is rational polyhedral. In fact, we prove more generally that this holds for any Mori dream space which admits a flag of Mori dream spaces satisfying a certain regularity condition. As a corollary, Okounkov bodies of effective line bundles over Schubert varieties are shown to be rational polyhedral. In particular, it follows that the global Okounkov body of a flag variety $G/B$ is rational polyhedral. As an application we show that the asymptotic behaviour of dimensions of weight spaces in section spaces of line bundles is given by the counting of lattice points in polytopes.Comment: A new and simpler definition of a good flag is introduced, and Bott-Samelson varieties are shown to admit such flag