Please answer the following question: Information: - In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projective variety which is as simple as possible. The subject has its origins in the classical birational geometry of surfaces studied by the Italian school, and is currently an active research area within algebraic geometry. - In algebraic geometry, a Fano variety, introduced in , is a complete variety "X" whose anticanonical bundle "K" is ample. In this definition, one could assume that "X" is smooth over a field, but the minimal model program has also led to the study of Fano varieties with various types of singularities, such as terminal or klt singularities. - 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 : 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 into 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 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... - In mathematics, the intermediate Jacobian of a compact Kahler manifold or Hodge structure is a complex torus that is a common generalization of the Jacobian variety of a curve and the Picard variety and the Albanese variety. It is obtained by putting a complex structure on the torus "H"(M,R)/"H"(M,Z) for "n" odd. There are several different natural ways to put a complex structure on this torus, giving several different sorts of intermediate Jacobians, including one due to and one due to . The ones constructed by Weil have natural polarizations if "M" is projective, and so are abelian varieties, while the ones constructed by Griffiths behave well under holomorphic deformations. - In algebraic geometry, a cubic threefold is a hypersurface of degree 3 in 4-dimensional projective space. Cubic threefolds are all unirational, but used intermediate Jacobians to show that non-singular cubic threefolds are not rational. The space of lines on a non-singular cubic 3-fold is a Fano surface. - In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold. - In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space "V". The cases when and are the real projective line and the real projective plane, respectively, where R denotes the field of real numbers, R denotes ordered pairs of real numbers, and R denotes ordered triplets of real numbers. - In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold "M" has "n" dimensions; then any submanifold of "M" of dimensions is a hypersurface. Equivalently, the codimension of a hypersurface is one. For example, the "n"-sphere in R is called a hypersphere. Hypersurfaces occur frequently in multivariable calculus as level sets. - In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension2. Because of Chow's theorem any compact complex manifold of dimension 2 and with Kodaira dimension 2 will actually be an algebraic surface, and in some sense most surfaces are in this class. - Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. - In algebraic geometry, the Kodaira dimension ("X") (or canonical dimension) measures the size of the canonical model of a projective variety "X". After reading the paragraphs above, we are interested in knowing the entity with which 'fano surface' exhibits the relationship of 'subclass of'. Find the answer from the choices below. Choices: - algebra - algebraic surface - area - calculus - curve - definition - dimension - general - geometry - hypersurface - kahler manifold - line - manifold - mathematics - model - natural - research - show - space - sphere - structure - subject - surface - theorem - thought - type - variety - vector space
Answer:
algebraic surface