- NSF Award Search: Award# - PECASE: Intersection Theory On Moduli Spaces
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We then use this to develop topological controls on the intersection number of two curves. We also prove an adjunction formula which gives a topological condition that will guarantee a curve in a given homotopy class is embedded, extending previous work of Hutchings [JEMS 4 —61].
- Arithmetic Intersection Theory.
- Igusa class polynomials, embeddings of quartic CM fields, and arithmetic intersection theory;
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- Topics in Algebraic Geometry: Intersection Theory on Moduli Spaces.
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We investigate controls on intersections of the projections of curves to the 3 —manifold and we present conditions that will guarantee the projection of a curve to the 3 —manifold is an embedding. Finally we consider an application concerning pseudoholomorphic curves in manifolds admitting a certain class of holomorphic open book decomposition and an application concerning the existence of generalized pseudoholomorphic curves, as introduced by Hofer [Geom.
Source Geom. Zentralblatt MATH identifier Keywords pseudoholomorphic curves symplectic field theory Floer homology intersection theory. Siefring, Richard. Intersection theory of punctured pseudoholomorphic curves. Between the constructions a , b , c there are a number of relations, of which the main ones are as follows:. Chern class with values in Chow rings, and in particular the Chern character.
It is simplest to determine the direct image homomorphism. Let be an irreducible subvariety; if , then , while if , then , where is the degree of over. By linearity, the definition is extended to cycles and classes of cycles. The inverse image homomorphism amounts to multiplication of cycles in accordance with.
The definition of multiplication of cycles is given in two stages. Let, initially, and be irreducible subvarieties in that intersect properly i.
NSF Award Search: Award# - PECASE: Intersection Theory On Moduli Spaces
Each component of the intersection is ascribed some positive integer , which is the local multiplicity of the intersection. There are several definitions of , for example, Serre's Tor-formula:. After this, one puts. The second stage is Chow's moving lemma: For any and on a quasi-projective variety there exists a cycle that is rationally equivalent to and that intersects properly with ; moreover, the rational equivalence class of is independent of.
The most interesting case is that of a projective variety ; applying the direct image functor to the structure morphism one obtains the mapping. In essence, the degree of a cycle is the number of points in its zero-dimensional component.
The composition of multiplication with the degree enables one to measure an intersection numerically. For example, if and have complementary dimensions, then one obtains the intersection index in algebraic geometry the intersection number of and.
Similarly, one obtains an intersection index for divisors :. For example, the Chow ring for the projective space is generated by the class of a hyperplane , where. Therefore, if are hypersurfaces of degrees , then Bezout's theorem.
The degree of a projective variety of dimension is defined as the intersection index of with a linear subspace of complementary dimension; if the varieties and intersect transversally, then the degree of is the product of the degrees of and. For properly-intersecting effective divisors , but in the general case this is not true. For example, for an exceptional curve cf. Exceptional subvariety on a surface,.
Other theories have many of the formal properties of the theory of Chow rings: cycles modulo algebraic or numerical equivalence, -theory, singular cohomology theory in the case , and -adic cohomology theory see also Weil cohomology. This leads to the axiomatic construction of intersection theory by putting each variety from some category into correspondence with a ring and homomorphisms and related by axioms of the type of projection or reduction-to-the-diagonal formulas see . The comparison of different intersection theories leads to useful results.
For example, in the complex case, the concept of a fundamental cycle enables one to define an intersection-theory homomorphism , which enables one to use transcendental methods. Comparison of -theory with Chow's theory leads to the Riemann—Roch—Grothendieck theorem cf.
Riemann—Roch theorem. An important part is played here by the intersection theory for monoidal transformations  , .
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Another application of intersection theory is related to the foundations of Schubert's geometric calculus . This branch of geometry can be regarded as the theory of Chow rings for various varieties that classify geometric objects: Grassmann manifolds, flag manifolds, etc. Fulton has defined Chow groups for singular varieties [a1].
A more refined intersection theory has been developed by Fulton and R. MacPherson: given cycles and on , it gives a well-defined element of [a2].
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A new development is the intersection theory for arithmetic varieties, i. Log in. Namespaces Page Discussion. Views View View source History.