Are there some upper bounds for the Picard number of a non-singular threefold? We know that in the surface case, we have $h^{1,1}=10\chi-c_1^2+2q$. Hence the picard number $\leq 10\chi-c_1^2+2q$. Is there a similar bound for 3-fold? Thank you!
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
A smooth Fano threefold $X$ has $\rho(X) \leq 10$. The only case with Picard number exactly $10$ is $X = Y\times\mathbb{P}^1$, where $Y$ is a Del Pezzo surface of degree one.
Iskovskih, V. A. (1977), "Fano threefolds. I", Math. USSR-Izv. 11 (3): 485–527.
Iskovskih, V. A. (1978), "Fano 3-folds II", Math Ussr Izv 12 (3): 469–506.
If $X$ is a weak Fano toric $3$-fold ($X$ is normal, Gorenstein, and $-K_X$ is nef and big) then $\rho(X)\leq 35$.
For a smooth complex projective variety $X$ we have $$\rho(X)\leq h^{1,1}(X) = b_2(X)-2\cdot h^{2,0}(X).$$ http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1981_4_14_3/ASENS_1981_4_14_3_303_0/ASENS_1981_4_14_3_303_0.pdf
