Constructions and moduli of surfaces of general type and related topics

In this thesis we treat two topics: the construction of minimal complex surfaces of general type with $p_g=q=2,3$ and an extension of Schur's concept of a representation group for projective representations to the setting of semi-projective representations. These are the contents of the two articles [AC22] and [AGK23], which are two joint works: the former with Fabrizio Catanese, the latter with Christian Gleissner and Julia Kotonski. The first part of the thesis is devoted to the treatment of the construction method for minimal surfaces of general type with $p_g=q$ developed together with Fabrizio Catanese in [AC22]. We give first a construction of minimal surfaces of general type with $p_g=q=2$, $K^2=5$ and Albanese map of degree 3, describing a unirational irreducible connected component of the Gieseker moduli space, which we show to be the only one with these invariants fulfilling a mild technical assumption (which we call Gorenstein Assumption) and whose general element $S$ has Albanese surface $Alb(S)$ containing no elliptic curve. We call it the component of "CHPP surfaces", since it contains the family constructed by Chen and Hacon in [CH06], and coincides with the one constructed by Penegini and Polizzi in [PePo13a]. Similarly, we construct a unirational irreducible connected component of the moduli space of minimal surfaces of general type with $p_g=q=2$, $K^2=6$ and Albanese map of degree 4, which we call the component of "PP4 surfaces" since it coincides with the irreducible one constructed by Penegini and Polizzi in [PePo14]. Furthermore, we answer a question posed by Chen and Hacon in [CH06] by constructing three families of surfaces with $p_g=q$ whose Tschirnhaus module has a kernel realization with quotient a nontrivial homogeneous bundle. Two families have $p_g=q=3$ (one of them is just a potential example since a computer script showing the existence is still missing), while the third one is a new family of surfaces with $p_g=q=2$, $K^2=6$ and Albanese map of degree 3. The latter, whose existence is showed in [CS22], yields a new irreducible component of the Gieseker moduli space, which we call the component of "AC3 surfaces". This is the first known component with these invariants, and moreover we show that it is unirational. We point out that we provide explicit and global equations for all the five families of surfaces we mentioned above. Finally, in the second and last part of the thesis we treat the content of the joint work [AGK23] with Christian Gleissner and Julia Kotonski. Here we study "semi-projective representations", i.e., homomorphisms of finite groups to the group of semi-projective transformations of finite dimensional vector spaces over an arbitrary field $K$. The main tool we use is "group cohomology", more precisely explicit computations involving cocycles. As our main result, we extend Schur's concept of "projective representation groups" [Sch04] to the semi-projective case under the assumption that $K$ is algebraically closed. Furthermore, a computer algorithm is given: it produces, for a given finite group, all "twisted representation groups" under trivial or conjugation actions on the field of complex numbers. In order to stress the relevance of the theory, we discuss two important applications, where semi-projective representations occur naturally. The first one reviews Isaacs' treatment in "Clifford theory for characters" [Isa81], namely the extension problem of invariant characters (over arbitrary fields) defined on normal subgroups. The second one is our original algebro-geometric motivation and deals with the problem to find linear parts of homeomorphisms and biholomorphisms between complex torus quotients. References: [AC22] Massimiliano Alessandro and Fabrizio Catanese. "On the components of the Main Stream of the moduli space of surfaces of general type with p_g=q=2". Preprint 2022 (arXiv:2212.14872v3). To appear in "Perspectives on four decades: Algebraic Geometry 1980-2020. In memory of Alberto Collino". Trends in Mathematics, Birkhäuser. [AGK23] Massimiliano Alessandro, Christian Gleissner and Julia Kotonski. "Semi-projective representations and twisted representation groups". Comm. Algebra 51 (2023), no. 10, 4471--4480. [CS22] Fabrizio Catanese and Edoardo Sernesi. "The Hesse pencil and polarizations of type (1,3) on abelian surfaces". Preprint 2022 (arXiv:2212.14877). To appear in "Perspectives on four decades: Algebraic Geometry 1980-2020. In memory of Alberto Collino". Trends in Mathematics, Birkhäuser. [CH06] Jungkai Alfred Chen and Christopher Derek Hacon. "A surface of general type with p_g=q=2 and K^2=5". Pacific J. Math. 223 (2006), no. 2, 219--228. [Isa81] Irving Martin Isaacs. "Extensions of group representations over arbitrary fields". J. Algebra 68 (1981), no. 1, 54--74. [PePo13a] Matteo Penegini and Francesco Polizzi. "On surfaces with p_g=q=2, K^2=5 and Albanese map of degree 3". Osaka J. Math. 50 (2013), no. 3, 643--686. [PePo14] Matteo Penegini and Francesco Polizzi. "A new family of surfaces with p_g=q=2 and K^2=6 whose Albanese map has degree 4". J. Lond. Math. Soc. (2) 90 (2014), no. 3, 741--762. [Sch04] Issai Schur. "Über die Darstellung der endlichen Gruppen durch gebrochen lineare Substitutionen". J. Reine Angew. Math. 127 (1904), 20--50 (German).