Fano surfaces are perhaps the simplest and most studied examples of irregular surfaces of general type that are not related to a product of two curves and are not a complete intersection of divisors in an Abelian variety.
The Fano surface S of a smooth cubic threefold F into P4 carries many remarkable geometric properties. The surface S is naturally embedded into the grassmannian of lines G(2,5) of P4. Let U be the restriction to S of the universal rank 2 bundle on G. We have the:
Tangent bundle Theorem (Fano, Clemens-Griffiths, Tyurin): The tangent bundle of S is isomorphic to U.
This is a quite interesting result because, a priori, there should be no link between these two bundles. It has many powerful applications. By example, one can recover the fact that the cotangent space of S is generated by global sections. This space of global 1-forms can be identified with the space of global sections of the tautological line bundle O(1) restricted to the cubic F and moreover:
Torelli-type Theorem : Let g' be the natural morphism from S to the grassmannian G(2,5) defined by the cotangent sheaf of S generated by its 5-dimensional space of global sections. Let F' be the union of the lines corresponding to g'(S). The threefold F' is isomorphic to F.
Thus knowing a Fano surface S, we can recover the threefold F. By the Tangent Bundle Theorem, we can also understand geometrically the invariants of S:
a) Recall that the second Chern number of a rank 2 vector bundle on a surface is the number of zeroes of a generic section. For a Fano surface S, a 1-form w defines also a hyperplane section {w=0} into P4 of the cubic F. The zeros of the generic w on S corresponds bijectively to the numbers of lines into the smooth cubic surface intersection of {w=0} and F, therefore we recover that the second Chern class of S equals 27.
b) Let w1, w2 be two 1-forms on S. The canonical divisor K on S associated to the canonical form w1 ∧ w2 parametrizes the lines on F that cut the plane P={w1=w2=0} into P4. Using w1 and w2 such that the intersection of P and F is the union of 3 lines, one can recover the fact that K2=45. Let us give some details of that computation: By a generic point of the cubic F goes 6 lines. Let s be a point of S and let Ls be the corresponding line on the cubic F. Let Cs be the divisor on S parametrizing lines that cut the line Ls. The self-intersection of Cs is equal to the intersection number of Cs and Ct for t a generic point. The intersection of Cs and Ct is the set of lines on F that cuts the disjoint lines Ls and Lt. Consider the linear span of Ls and Lt : it is an hyperplane into P4 that cuts F into a smooth cubic surface. By well known results on a cubic surface, the number of lines that cuts two disjoints lines is 5, thus we get (Cs) 2 =CsCt=5. As K is numerically equivalent to 3Cs, we obtain K 2 =45.
c) The natural composite map: S -> G(2,5) -> P9 is the canonical map of S. It is an embedding.
Fano, G. (1904), "Sul sistema ∞2 di rette contenuto in une varietà cubica generale dello spazio a quattro dimensioni", Atti R. Accad. Sci. Torino, 39: 778–792
Murre, J. P. (1972), "Algebraic equivalence modulo rational equivalence on a cubic threefold", Compositio Mathematica, 25: 161–206, ISSN 0010-437X, MR 0352088
October 27, 2023
fano, surface, fano, varieties, dimension, pezzo, surface, projective, plane, over, field, with, elements, fano, plane, algebraic, geometry, surface, general, type, particular, fano, variety, whose, points, index, lines, singular, cubic, threefold, they, were,. For Fano varieties of dimension 2 see del Pezzo surface For the projective plane over the field with 2 elements see Fano plane In algebraic geometry a Fano surface is a surface of general type in particular not a Fano variety whose points index the lines on a non singular cubic threefold They were first studied by Fano 1904 Hodge diamond 155102510551Fano surfaces are perhaps the simplest and most studied examples of irregular surfaces of general type that are not related to a product of two curves and are not a complete intersection of divisors in an Abelian variety The Fano surface S of a smooth cubic threefold F into P4 carries many remarkable geometric properties The surface S is naturally embedded into the grassmannian of lines G 2 5 of P4 Let U be the restriction to S of the universal rank 2 bundle on G We have the Tangent bundle Theorem Fano Clemens Griffiths Tyurin The tangent bundle of S is isomorphic to U This is a quite interesting result because a priori there should be no link between these two bundles It has many powerful applications By example one can recover the fact that the cotangent space of S is generated by global sections This space of global 1 forms can be identified with the space of global sections of the tautological line bundle O 1 restricted to the cubic F and moreover Torelli type Theorem Let g be the natural morphism from S to the grassmannian G 2 5 defined by the cotangent sheaf of S generated by its 5 dimensional space of global sections Let F be the union of the lines corresponding to g S The threefold F is isomorphic to F Thus knowing a Fano surface S we can recover the threefold F By the Tangent Bundle Theorem we can also understand geometrically the invariants of S a Recall that the second Chern number of a rank 2 vector bundle on a surface is the number of zeroes of a generic section For a Fano surface S a 1 form w defines also a hyperplane section w 0 into P4 of the cubic F The zeros of the generic w on S corresponds bijectively to the numbers of lines into the smooth cubic surface intersection of w 0 and F therefore we recover that the second Chern class of S equals 27 b Let w1 w2 be two 1 forms on S The canonical divisor K on S associated to the canonical form w1 w2 parametrizes the lines on F that cut the plane P w1 w2 0 into P4 Using w1 and w2 such that the intersection of P and F is the union of 3 lines one can recover the fact that K2 45 Let us give some details of that computation By a generic point of the cubic F goes 6 lines Let s be a point of S and let Ls be the corresponding line on the cubic F Let Cs be the divisor on S parametrizing lines that cut the line Ls The self intersection of Cs is equal to the intersection number of Cs and Ct for t a generic point The intersection of Cs and Ct is the set of lines on F that cuts the disjoint lines Ls and Lt Consider the linear span of Ls and Lt it is an hyperplane into P4 that cuts F into a smooth cubic surface By well known results on a cubic surface the number of lines that cuts two disjoints lines is 5 thus we get Cs 2 Cs Ct 5 As K is numerically equivalent to 3Cs we obtain K 2 45 c The natural composite map S gt G 2 5 gt P9 is the canonical map of S It is an embedding See also editHodge theoryReferences editBombieri Enrico Swinnerton Dyer H P F 1967 On the local zeta function of a cubic threefold Ann Scuola Norm Sup Pisa 3 21 1 29 MR 0212019 Clemens C Herbert Griffiths Phillip A 1972 The intermediate Jacobian of the cubic threefold Annals of Mathematics Second Series 95 2 281 356 CiteSeerX 10 1 1 401 4550 doi 10 2307 1970801 ISSN 0003 486X JSTOR 1970801 MR 0302652 Fano G 1904 Sul sistema 2 di rette contenuto in une varieta cubica generale dello spazio a quattro dimensioni Atti R Accad Sci Torino 39 778 792 Kulikov Vik S 2001 1994 Fano surface Encyclopedia of Mathematics EMS Press Murre J P 1972 Algebraic equivalence modulo rational equivalence on a cubic threefold Compositio Mathematica 25 161 206 ISSN 0010 437X MR 0352088 Retrieved from https en wikipedia org w index php title Fano surface amp oldid 1085250772, wikipedia, wiki, book, books, library,
