Algebraic and complex surfaces abelian surfaces (κ = 0) Two-dimensional abelian varieties. algebraic surfaces Barlow surfaces General type, simply connected. Barth surfaces Surfaces of degrees 6 and 10 with many nodes. Beauville surfaces General type bielliptic surfaces (κ = 0) Same as hyperelliptic surfaces. Bordiga surfaces A degree-6 embedding of the projective plane into P4 defined by the quartics through 10 points in general position. Burniat surfaces General type Campedelli surfaces General type Castelnuovo surfaces General type Catanese surfaces General type Cayley surface Rational. A cubic surface with 4 nodes. Châtelet surfaces Rational class VII surfaces κ = −∞, non-algebraic. Clebsch surface Rational. The surface Σxi = Σxi3 = 0 in P4. Coble surfaces Rational cubic surfaces Rational. Del Pezzo surfaces Rational. Anticanonical divisor is ample, for example P2 blown up in at most 8 points. Dolgachev surfaces Elliptic. elliptic surfaces Surfaces with an elliptic fibration. Endrass surface A surface of degree 8 with 168 nodes Enneper surface Enoki surface Class VII Enriques surfaces (κ = 0) exceptional surfaces: Picard number has the maximal possible value h1,1. fake projective plane general type, found by Mumford, same Betti numbers as projective plane. Fano surface of lines on a non-singular 3-fold. It can also mean del Pezzo surface. Fermat surface of degree d: Solutions of wd + xd + yd + zd = 0 in P3. general type κ = 2 generalized Raynaud surface in positive characteristic Godeaux surfaces (general type) Hilbert modular surfaces Hirzebruch surfaces Rational ruled surfaces. Hopf surfaces κ = −∞, non-algebraic, class VII Horikawa surfaces general type Horrocks–Mumford surfaces. These are certain abelian surfaces of degree 10 in P4, given as zero sets of sections of the rank 2 Horrocks–Mumford bundle. Humbert surfaces These are certain surfaces in quotients of the Siegel upper half-space of genus 2. hyperelliptic surfaces κ = 0, same as bielliptic surfaces. Inoue surfaces κ = −∞, class VII,b2 = 0. (Several quite different families were also found by Inoue, and are also sometimes called Inoue surfaces.) Inoue-Hirzebruch surfaces κ = −∞, non-algebraic, type VII, b2>0. K3 surfaces κ = 0, supersingular K3 surface. Kähler surfaces complex surfaces with a Kähler metric, which exists if and only if the first Betti number b1 is even. Kato surface Class VII Klein icosahedral surface The Clebsch cubic surface or its blowup in 10 points. Kodaira surfaces κ = 0, non-algebraic Kummer surfaces κ = 0, special sorts of K3 surfaces. minimal surfaces Surfaces with no rational −1 curves. (They have no connection with minimal surfaces in differential geometry.) Mumford surface A "fake projective plane" non-classical Enriques surface Only in characteristic 2. numerical Campedelli surfaces surfaces of general type with the same Hodge numbers as a Campedelli surface. numerical Godeaux surfaces surfaces of general type with the same Hodge numbers as a Godeaux surface. Picard modular surface Plücker surface Birational to Kummer surface projective plane Rational properly elliptic surfaces κ = 1, elliptic surfaces of genus ≥2. quadric surfaces Rational, isomorphic to P1 × P1. quartic surfaces Nonsingular ones are K3s. quasi Enriques surface These only exist in characteristic 2. quasi elliptic surface Only in characteristic p > 0. quasi-hyperelliptic surface quotient surfaces: Quotients of surfaces by finite groups. Examples: Kummer, Godeaux, Hopf, Inoue surfaces. rational surfaces κ = −∞, birational to projective plane Raynaud surface in positive characteristic Reye congruence A special sort of Enriques surface. κ=0. Roman surface ruled surfaces κ = −∞ Sarti surface A degree-12 surface in P3 with 600 nodes. Segre surface An intersection of two quadrics, isomorphic to the projective plane blown up in 5 points. Steiner surface A surface in P4 with singularities which is birational to the projective plane. surface of general type κ = 2. Tetrahedroid A special Kummer surface. Togliatti surfaces, degree-5 surfaces in P3 with 31 nodes. unirational surfaces Castelnuovo proved these are all rational in characteristic 0. Veronese surface An embedding of the projective plane into P5. Wave surface A special Kummer surface. Weddle surface κ = 0, birational to Kummer surface. White surface Rational. Zariski surfaces (only in characteristic p > 0): There is a purely inseparable dominant rational map of degree p from the projective plane to the surface. |

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