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Hibert Scheme and Gromov-Witten Invariant

Amini, Kamyar | 2019

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 52860 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Safdari, Mohammad; Setayesh, Iman
  7. Abstract:
  8. The aim of these notes is to describe an exciting chapter in the recent development of quantum cohomology. Guided by ideas from physics, a remarkable structure on the solutions of certain rational enumerative geometry problems has been found: the solutions are coefficients in the multiplication table of a quantum cohomology ring. Associativity of the ring yields non-trivial relations among the enumerative solutions. In many cases, these relations suffice to solve the enumerative problem. For example, let $N_d$ be the number of degree $d$, rational plane curves passing through $3d$ − $1$ general points in $\mathbb{P}^2$. Since there is a unique line passing through 2 points, $N_1 = 1$. The quantum cohomology ring of $\mathbb{P}^2$ yields the following beautiful associativity relation determining all $N_d$ for $d$ $\geg$ $2$:
    N_d=∑_(d_1+d_2=d)▒〖N_(d_1 ) N_(d_2 ) [d_1^2 d_1^2 (■(3d-4@3d_1-2))-d_1^3 d_2^1 (■(3d-4@3d_1-1))] 〗
    Similar enumerative formulas are valid on other homogeneous varieties. Viewed from classical enumerative geometry, the quantum ring structure is a complete surprise. The path to quantum cohomology presented here follows the work of Kontsevich and Manin. The approach is algebro-geometric and involves the construction and geometry of a natural compactification of the moduli space of maps. The large and exciting conjectural parts of the subject of quantum cohomology are avoided here. We focus on a part of the story where the proofs are complete. We also make many assumptions that are not strictly necessary, but which simplify the presentation
  9. Keywords:
  10. Moduli Space ; Algebraic Geometry ; Quantum Cohomology ; Hibert Scheme ; Gromov-Witten Invariants

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