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By Paul Hacking, Radu Laza, Dragos Oprea, Gilberto Bini, Martí Lahoz, Emanuele Macrí, Paolo Stellari

This e-book focusses on a wide category of gadgets in moduli thought and offers diversified views from which compactifications of moduli areas can be investigated.

Three contributions provide an perception on specific elements of moduli difficulties. within the first of them, quite a few how one can build and compactify moduli areas are offered. within the moment, a few questions about the boundary of moduli areas of surfaces are addressed. ultimately, the speculation of reliable quotients is defined, which yields significant compactifications of moduli areas of maps.

either complex graduate scholars and researchers in algebraic geometry will locate this ebook a helpful read.

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An ) = Proj k[X0 , . . , Xn ], for j = i. We where the grading of the polynomial ring is given by deg Xi = ai . The sheaf OP (1) = OP (H) is a rank-one reflexive sheaf corresponding to a Weil divisor class H. The global sections of OP (n) = OP (nH) are the homogeneous polynomials of (weighted) degree n. The divisor class group Cl(P) is isomorphic to Z, generated by H. The divisor H is Q-Cartier, and satisfies H n = 1/(a0 · · · an ). 1. Moduli spaces of surfaces of general type 43 The canonical divisor class KP is given by KP = −(a0 + a1 + · · · + an )H.

Grushevsky, K. Hulek, and R. Laza. Extending the Prym map to toroidal compactifications of the moduli space of abelian varieties. Preprint, 2014. [29] S. Casalaina-Martin, D. Jensen, and R. Laza. The geometry of the ball quotient model of the moduli space of genus four curves. In Compact Moduli Spaces and Vector Bundles, volume 564 of Contemp. , pages 107–136. Amer. Math. , 2012. [30] S. Casalaina-Martin, D. Jensen, and R. Laza. Log canonical models and variation of GIT for genus 4 canonical curves.

J. Amer. Math. , 23(2):405–468, 2010. [22] C. Borcea. K3 surfaces with involution and mirror pairs of Calabi–Yau manifolds. In Mirror Symmetry, II, volume 1 of AMS/IP Stud. Adv. , pages 717–743. Amer. Math. , Providence, RI, 1997. E. Borcherds, L. Katzarkov, T. I. Shepherd-Barron. Families of K3 surfaces. J. , 7(1):183–193, 1998. [24] A. Borel. Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem. J. Differential Geometry, 6:543–560, 1972. S. C. Spencer on their sixtieth birthdays.

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