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

This publication focusses on a wide type of gadgets in moduli idea and offers diverse views from which compactifications of moduli areas could be investigated.

Three contributions supply an perception on specific features 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 speculation of reliable quotients is defined, which yields significant compactifications of moduli areas of maps.

either complicated graduate scholars and researchers in algebraic geometry will locate this publication a useful read.

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37] P. Deligne and D. Mumford. The irreducibility of the space of curves of given ´ genus. Inst. Hautes Etudes Sci. Publ. , 36:75–109, 1969. V. Dolgachev. Lectures on Invariant Theory, volume 296 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2003. V. Dolgachev and Y. Hu. Variation of geometric invariant theory quotients. ´ Inst. Hautes Etudes Sci. Publ. , 87:5–56, 1998. V. Dolgachev and S. Kond¯o. Moduli of K3 surfaces and complex ball quotients. In Arithmetic and Geometry Around Hypergeometric Functions, volume 260 of Progr.

In our case KX = OX (d − 4), d − 4 > 0, 46 Chapter 2. Compact Moduli of Surfaces and Vector Bundles and KX deforms so, OX (1) deforms as required. 4 Expected dimension We can compute the expected dimension of M using the Hirzebruch–Riemann– Roch formula. Let X be a smooth projective surface. Write c1 = c1 (TX ) = −KX and (−1)i dimR H i (X, R). c2 = c2 (TX ) = e(X) = i For F a vector bundle on X the Hirzebruch–Riemann–Roch formula states that χ(F ) = (ch(F ) · td(X))2 , where ch(F ) = rk(F ) + c1 (F ) + 1 c1 (F )2 − 2c2 (F ) 2 is the Chern character, and 1 1 td(X) = 1 + c1 + (c21 + c2 ) 2 12 is the Todd class.

6]. Let n ∈ N and a0 , . . , an ∈ N. We write P = P(a0 , . . , an ) for the weighted projective space P(a0 , . . , an ) = (An+1 \ {0})/Gm, Gm λ : (X0 , . . , Xn ) −→ (λa0 X0 , . . , λan Xn ). We always assume that gcd(a0 , . . , ai , . . , an ) = 1 for all i. Then P(a0 , . . , an ) is a normal projective variety covered by aﬃne charts (Xi = 0) = An / a1i (a0 , . . , ai , . . , an ), a /ai where the aﬃne orbifold coordinates are given by xji = Xj /Xi j have P(a0 , . . , an ) = Proj k[X0 , .