A Lower Bound for the Sectional Genus of Quasi-Polarized Surfaces
Tóm tắt
Let (X,L) be a quasi-polarized variety, i.e. X is a smooth projective variety over the complex numbers
$$\mathbb{C}$$
and L is a nef and big divisor on X. Then we conjecture that g(L) = q(X), whereg(L) is the sectional genus of L and
$$q(X) = \dim H^1 (\mathcal{O}_X )$$
. In this paper, we treat the case
$$\dim X = 2$$
. First we prove that this conjecture is true for
$$\kappa (X) \leqslant 1$$
, and we classify (X,L) withg(L)=q(X), where
$$\kappa (X)$$
is the Kodaira dimension of X. Next we study some special cases of
$$\kappa (X) = 2$$
.
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