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Tautological Ring
In algebraic geometry, the tautological ring is the subring of the Chow ring of the moduli space of curves generated by tautological classes. These are classes obtained from 1 by pushforward along various morphisms described below. The tautological cohomology ring is the image of the tautological ring under the cycle map (from the Chow ring to the cohomology ring). Definition Let \overline_ be the moduli stack of stable marked curves (C;x_1,\ldots,x_n), such that * ''C'' is a complex curve of arithmetic genus ''g'' whose only singularities are nodes, * the ''n'' points ''x''1, ..., ''x''''n'' are distinct smooth points of ''C'', * the marked curve is stable, namely its automorphism group (leaving marked points invariant) is finite. The last condition requires 2g-2+n>0 in other words (''g'',''n'') is not among (0,0), (0,1), (0,2), (1,0). The stack \overline_ then has dimension 3g-3+n. Besides permutations of the marked points, the following morphisms between these moduli stac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
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. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chow Ring
In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups (compare Poincaré duality) and have a multiplication called the intersection product. The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general. Rational equivalence and Chow groups For what follows, define a variety over a field k to be an integral scheme of finite type over k. For any scheme X of finite type over k, an algebraic cycle on X means a finite linear combination of subvarieties of X with integer coefficients. (Here and below, subvarieties are understood to b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moduli Space Of Curves
In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. One also distinguishes between fine and coarse moduli spaces for the same moduli problem. The most basic problem is that of moduli of smooth complete curves of a fixed genus. Over the field of complex numbers these correspond precisely to compact Riemann surfaces of the given genus, for which Bernhard Riemann proved the first results about moduli spaces, in particular their dimensions ("number of parameters on which the complex structure depends"). Moduli stacks of stable curves The moduli stack \mathcal_ classifies families of smooth projective curves, together with their isomorphisms. Whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moduli Of Algebraic Curves
In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. One also distinguishes between fine and coarse moduli spaces for the same moduli problem. The most basic problem is that of moduli of smooth complete curves of a fixed genus. Over the field of complex numbers these correspond precisely to compact Riemann surfaces of the given genus, for which Bernhard Riemann proved the first results about moduli spaces, in particular their dimensions ("number of parameters on which the complex structure depends"). Moduli stacks of stable curves The moduli stack \mathcal_ classifies families of smooth projective curves, together with their isomorphisms. When g > 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ELSV Formula
In mathematics, the ELSV formula, named after its four authors , , Michael Shapiro, Alek Vainshtein, is an equality between a Hurwitz number (counting ramified coverings of the sphere) and an integral over the moduli space of stable curves. Several fundamental results in the intersection theory of moduli spaces of curves can be deduced from the ELSV formula, including the Witten conjecture, the Virasoro constraints, and the \lambda_g-conjecture. It is generalized by the Gopakumar–Mariño–Vafa formula. The formula Define the ''Hurwitz number'' : h_ as the number of ramified coverings of the complex projective line (Riemann sphere, \mathbb^1(\C)) that are connected curves of genus ''g'', with ''n'' numbered preimages of the point at infinity having multiplicities k_1, \dots, k_n and ''m'' more simple branch points. Here if a covering has a nontrivial automorphism group ''G'' it should be counted with weight 1/, G, . The ELSV formula then reads : h_ = \dfrac \prod_^n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hodge Bundle
In mathematics, the Hodge bundle, named after W. V. D. Hodge, appears in the study of families of curves, where it provides an invariant in the moduli theory of algebraic curves. Furthermore, it has applications to the theory of modular forms on reductive algebraic groups and string theory. Definition Let \mathcal_g be the moduli space of algebraic curves of genus ''g'' curves over some scheme. The Hodge bundle \Lambda_g is a vector bundleHere, "vector bundle" in the sense of quasi-coherent sheaf on an algebraic stack on \mathcal_g whose fiber at a point ''C'' in \mathcal_g is the space of holomorphic differentials on the curve ''C''. To define the Hodge bundle, let \pi\colon \mathcal_g\rightarrow\mathcal_g be the universal algebraic curve of genus ''g'' and let \omega_g be its relative dualizing sheaf. The Hodge bundle is the pushforward The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Witten's Conjecture
In algebraic geometry, the Witten conjecture is a conjecture about intersection numbers of stable classes on the moduli space of curves, introduced by Edward Witten in the paper , and generalized in . Witten's original conjecture was proved by Maxim Kontsevich in the paper . Witten's motivation for the conjecture was that two different models of 2-dimensional quantum gravity should have the same partition function. The partition function for one of these models can be described in terms of intersection numbers on the moduli stack of algebraic curves, and the partition function for the other is the logarithm of the τ-function of the KdV hierarchy. Identifying these partition functions gives Witten's conjecture that a certain generating function formed from intersection numbers should satisfy the differential equations of the KdV hierarchy. Statement Suppose that ''M''''g'',''n'' is the moduli stack of compact Riemann surfaces of genus ''g'' with ''n'' distinct marked points ''x' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
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. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
