By Paul Hacking, Radu Laza, Dragos Oprea, Gilberto Bini, Martí Lahoz, Emanuele Macrí, Paolo Stellari
This e-book focusses on a wide type of gadgets in moduli idea and offers assorted views from which compactifications of moduli areas should be investigated.
Three contributions provide an perception on specific points of moduli difficulties. within the first of them, numerous how you can build and compactify moduli areas are provided. within the moment, a few questions about the boundary of moduli areas of surfaces are addressed. eventually, the idea of sturdy quotients is defined, which yields significant compactifications of moduli areas of maps.
either complex graduate scholars and researchers in algebraic geometry will locate this publication a invaluable read.
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Extra resources for Compactifying Moduli Spaces
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The original construction of Mg does not involve GIT. , ﬂat proper morphisms X/S). , local patches of M) of the objects under consideration. From this point of view, the main issue is to select a class of objects so that the moduli stack is proper and separated. , smooth varieties of general type (with ample KX ) vs. GIT semistable varieties (say, with respect to the embedding given by nKX ), but not both at the same time. By the valuative criteria, deﬁning a proper and separated moduli stack is essentially equivalent to asking that a 1-parameter family X∗ /Δ∗ over the punctured disk has a unique limit with respect to the given moduli functor.