Enriques surface

The plurigenera P_n are 1 if n is even and 0 if n is odd. The fundamental group has order 2. The second cohomology group H²(X, Z) is isomorphic to the sum of the unique even unimodular lattice II_1,9 of dimension 10 and signature -8 and a group of order 2.

Hodge diamond:

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| 0 | 10 | 0

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Marked Enriques surfaces form a connected 10-dimensional family, which {{harvtxt|Kondo|1994}} showed is rational.

In characteristic 2 there are some new families of Enriques surfaces,

sometimes called quasi Enriques surfaces or non-classical Enriques surfaces or (super)singular Enriques surfaces. (The term "singular" does not mean that the surface has singularities, but means that the surface is "special" in some way.)

In characteristic 2 the definition of Enriques surfaces is modified: they are defined to be minimal surfaces whose canonical class K is numerically equivalent to 0 and whose second Betti number is 10. (In characteristics other than 2 this is equivalent to the usual definition.) There are now 3 families of Enriques surfaces:

• Classical: dim(H¹(O)) = 0. This implies 2K = 0 but K is nonzero, and Pic^τ is Z/2Z. The surface is a quotient of a reduced singular Gorenstein surface by the group scheme μ₂.
• Singular: dim(H¹(O)) = 1 and is acted on non-trivially by the Frobenius endomorphism. This implies K = 0, and Pic^τ is μ₂. The surface is a quotient of a K3 surface by the group scheme Z/2Z.
• Supersingular: dim(H¹(O)) = 1 and is acted on trivially by the Frobenius endomorphism. This implies K = 0, and Pic^τ is α₂. The surface is a quotient of a reduced singular Gorenstein surface by the group scheme α₂.

All Enriques surfaces are elliptic or quasi elliptic.

• A Reye congruence is the family of lines contained in at least 2 quadrics of a given 3-dimensional linear system of quadrics in P³. If the linear system is generic then the Reye congruence is an Enriques surface. These were found by {{harvtxt|Reye|1882}}, and may be the earliest examples of Enriques surfaces.
• Take a surface of degree 6 in 3 dimensional projective space with double lines along the edges of a tetrahedron, such as

::$w^2x^2y^2\; +\; w^2x^2z^2\; +\; w^2y^2z^2\; +\; x^2y^2z^2\; +\; wxyzQ(w,x,y,z)\; =\; 0$

:for some general homogeneous polynomial Q of degree 2. Then its normalization is an Enriques surface. This is the family of examples found by {{harvtxt|Enriques|1896}}.

• The quotient of a K3 surface by a fixed point free involution is an Enriques surface, and all Enriques surfaces in characteristic other than 2 can be constructed like this. For example, if S is the K3 surface w⁴ + x⁴ + y⁴ + z⁴ = 0 and T is the order 4 automorphism taking (w,x,y,z) to (w,ix,–y,–iz) then T² has eight fixed points. Blowing up these eight points and taking the quotient by T² gives a K3 surface with a fixed-point-free involution T, and the quotient of this by T is an Enriques surface. Alternatively, the Enriques surface can be constructed by taking the quotient of the original surface by the order 4 automorphism T and resolving the eight singular points of the quotient. Another example is given by taking the intersection of 3 quadrics of the form P_i(u,v,w) + Q_i(x,y,z) = 0 and taking the quotient by the involution taking (u:v:w:x:y:z) to (–x:–y:–z:u:v:w). For generic quadrics this involution is a fixed-point-free involution of a K3 surface so the quotient is an Enriques surface.

• List of algebraic surfaces
• Enriques–Kodaira classification
• Supersingular variety

• {{citation|first=Michael|last=Artin|authorlink = Michael Artin|title=On Enriques surfaces|publisher=Harvard|series = PhD thesis|year=1960}}
• Compact Complex Surfaces by Wolf P. Barth, Klaus Hulek, Chris A.M. Peters, Antonius Van de Ven {{ISBN|3-540-00832-2}} This is the standard reference book for compact complex surfaces.
• {{Citation | last1=Bombieri | first1=Enrico | author1-link=Enrico Bombieri | last2=Mumford | first2=David | author2-link=David Mumford | title=Enriques' classification of surfaces in char. p. III. | doi=10.1007/BF01390138 | mr=0491720 | year=1976 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=35 | issue=1 | pages=197–232| bibcode=1976InMat..35..197B | s2cid=122816845 | url=https://dash.harvard.edu/bitstream/handle/1/3612776/Mumford_EnriquesClassIII.pdf?sequence=3 }}
• {{citation|first=G.|last= Castelnuovo|title= Sulle superficie di genere zero|journal= Mem. Delle Soc. Ital. Delle Scienze |series=Série III|volume= 10 |year=1895|pages= 103–123}}
• {{Citation | last1=Cossec | first1=François R. | last2=Dolgachev | first2=Igor V. | title=Enriques surfaces. I | publisher=Birkhäuser Boston | location=Boston | series=Progress in Mathematics | isbn=978-0-8176-3417-9 | mr=986969 | year=1989 | volume=76}}
• {{Citation | last1=Dolgachev | first1=Igor V. | title=A brief introduction to Enriques surfaces|url=http://www.math.lsa.umich.edu/~idolga/Kyoto13.pdf|year=2016}}
• {{Citation|last=Enriques | first=Federigo|title=Introduzione alla geometria sopra le superficie algebriche.|year=1896|journal=Mem. Soc. Ital. Delle Scienze| volume=10|pages= 1–81}}
• {{Citation | last1=Enriques | first1=Federigo | title=Le Superficie Algebriche | url=https://www.afsu.it/wp-content/uploads/2020/08/F.-Enriques-1949.-Le-superficie-algebriche-Zanichelli-1949.pdf | publisher=Nicola Zanichelli, Bologna | mr=0031770 | year=1949 }}
• {{citation|last=Kondo|first= Shigeyuki |title=The rationality of the moduli space of Enriques surfaces|journal= Compositio Mathematica |volume= 91 |year=1994|issue= 2|pages= 159–173}}
• {{citation|first=T.|last=Reye|title=Die Geometrie der Lage|year =1882|publisher=Baumgärtnerś Buchhandlung|place=Leipzig|url=https://archive.org/details/diegeometrieder01reyegoog}}

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Category:Complex surfaces

Category:Birational geometry

Category:Algebraic surfaces