Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A degeneration formula of Gromov-Witten invariants with respect to a
  curve class for degenerations from blow-ups   In two very detailed, technical, and fundamental works, Jun Li constructed a
theory of Gromov-Witten invariants for a singular scheme of the gluing form
$Y_1\cup_D Y_2$ that arises from a degeneration $W/{\Bbb A}^1$ and a theory of
relative Gromov-Witten invariants for a codimension-1 relative pair $(Y,D)$. As
a summit, he derived a degeneration formula that relates a finite summation of
the usual Gromov-Witten invariants of a general smooth fiber $W_t$ of $W/{\Bbb
A}^1$ to the Gromov-Witten invariants of the singular fiber $W_0=Y_1\cup_D Y_2$
via gluing the relative pairs $(Y_1,D)$ and $(Y_2,D)$. The finite sum mentioned
above depends on a relative ample line bundle $H$ on $W/{\Bbb A}^1$. His theory
has already applications to string theory and mathematics alike. For other new
applications of Jun Li's theory, one needs a refined degeneration formula that
depends on a curve class $\beta$ in $A_{\ast}(W_t)$ or $H_2(W_t;{\Bbb Z})$,
rather than on the line bundle $H$. Some monodromy effect has to be taken care
of to deal with this. For the simple but useful case of a degeneration $W/{\Bbb
A}^1$ that arises from blowing up a trivial family $X\times{\Bbb A}^1$, we
explain how the details of Jun Li's work can be employed to reach such a
desired degeneration formula. The related set $\Omega_{(g,k;\beta)}$ of
admissible triples adapted to $(g,k;\beta)$ that appears in the formula can be
obtained via an analysis on the intersection numbers of relevant cycles and a
study of Mori cones that appear in the problem. This set is intrinsically
determined by $(g,k;\beta)$ and the normal bundle ${\cal N}_{Z/X}$ of the
smooth subscheme $Z$ in $X$ to be blown up.

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The categories of this paper are: mtrl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: chao-dyn
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The categories of this paper are: dg-ga
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The categories of this paper are: q-alg
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The categories of this paper are: econ
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: acc-phys
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The categories of this paper are: supr-con
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The categories of this paper are: solv-int