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ISBN-10: 0817644563

ISBN-13: 9780817644567

ISBN-10: 0817644954

ISBN-13: 9780817644956

This ebook is an easy advent to strong maps and quantum cohomology, beginning with an creation to solid pointed curves, and culminating with an explanation of the associativity of the quantum product. the point of view is generally that of enumerative geometry, and the pink thread of the exposition is the matter of counting rational aircraft curves. Kontsevich's formulation is first and foremost tested within the framework of classical enumerative geometry, then as a press release approximately reconstruction for Gromov–Witten invariants, and eventually, utilizing producing features, as a different case of the associativity of the quantum product.

Emphasis is given in the course of the exposition to examples, heuristic discussions, and straightforward purposes of the elemental instruments to top exhibit the instinct in the back of the topic. The ebook demystifies those new quantum options by way of displaying how they healthy into classical algebraic geometry.

Some familiarity with easy algebraic geometry and simple intersection thought is thought. each one bankruptcy concludes with a few ancient reviews and an overview of key subject matters and issues as a advisor for additional learn, through a set of routines that supplement the fabric coated and make stronger computational talents. As such, the booklet is perfect for self-study, as a textual content for a mini-course in quantum cohomology, or as a unique issues textual content in a regular path in intersection conception. The publication will end up both invaluable to graduate scholars within the school room surroundings as to researchers in geometry and physics who desire to know about the subject.

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Additional info for An Invitation to Quantum Cohomology: Kontsevich’s Formula for Rational Plane Curves

Example text

Pn) and an arbitrary point q E C, we are going to describe a canonical way to produce a stable {n + l)-pointed curve. If q is not a special point, then we just set pn-^i • = q and we have an (^ + l)-pointed curve that is obviously stable. If <7 is a special point of C, we may initially set pn+i = q; but then this (^ + l)-pointed curve is not stable. What we claim is that there is a canonical way of stabilizing a pointed curve of this type. We have already seen examples in the previous section hinting at how this must be done: to preserve continuity in the stabilization process, the limit of the stabilization must be the stabilization of the limit.

9, where we will see that the stratum in Mo,^ corresponding to curves with 8 nodes is locally a product of the moduli spaces of its twigs. 2 Example. Here is a drawing of the stratification of Mo,6- The six marked points have not been assigned names, but the number next to each figure indicates how many ways there are to label the given configuration: Each number appears as a multinomial coefficient, divided by the number of symmetries of the configuration. ) /^ because the configuration is symmetric in the middle.

13. Show that a coarse moduli space is unique up to unique isomorphism, if it exists. 9 is enough to guarantee this. 14. The subset classifier. Consider the moduli problem of classifying subsets. A family over a set B is defined to be a subset of B, and two subsets are considered equivalent when equal. Show that the inclusion {true} n {true, false} is a universal family. 15. Classifying finite sets. Counting is one of the most fundamental aspects of mathematics, and in this book we are very much concerned with counting.

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An Invitation to Quantum Cohomology: Kontsevich’s Formula for Rational Plane Curves by (auth.)


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