Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rational curves on generic quintic threefolds   Let $X_0$ be a generic quintic threefold in projective space $\mathbf P^4$
over complex numbers and $C_0$ be an irreducible rational curve on $X_0$. Let
$$c_0: \mathbf P^1\to C_0\subset X_0$$ be its normalization. In this paper, we
show (1) $c_0$ must be an immersion, i.e. the differential $(c_0)_\ast:
T_{t}\mathbf P^1 \to T_{c_0(t)} X_0$ is injective at each $t\in \mathbf P^1$,
(2) the normal bundle of $c_0$ satisfies $$H^1(N_{c_0/X_0})=0.$$

9
The categories of this paper are: nlin
28
The categories of this paper are: alg-geom
8
The categories of this paper are: hep-ex
27
The categories of this paper are: comp-gas
22
The categories of this paper are: chao-dyn
25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
29
The categories of this paper are: funct-an
19
The categories of this paper are: eess
1
The categories of this paper are: cs
26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
17
The categories of this paper are: q-fin
11
The categories of this paper are: hep-lat