7 added 5 characters in body

In the general case when $n=\dim X$ is arbitrary Kollár proves an analogue for minimal varieties of general type, although the bound is much worse than in the surface case. In the situation at hand it implies that $\vert ((n+3)!)K_X\vert$ is basepoint-free (actually one can write $(n+2)(n+2)!$ instead of $(n+3)!$, but this is shorter).

See: Kollár, János Effective base point freeness. Math. Ann. 296 (1993), no. 4, 595–605.

A closely related problem is Fujita's conjecture, which states that if $L$ is an ample divisor on $X$, then $|K_X+mL|$ is basepoint-free if $m\geq \dim X+1$ and very ample if $m\geq \dim X +2$.

There are many related results, but the conjecture is still open.

Angehrn and Siu prove that if $L$ is ample, then $|mL+K_X|$ is base point free if $m\geq 1/2(n^2+n+2)$\frac 12(n^2+n+2)$. Angehrn, Urban; Siu, Yum Tong Effective freeness and point separation for adjoint bundles. Invent. Math. 122 (1995), no. 2, 291–308. 32J25 (14C20 32L10) Helmke also has some related results: Helmke, Stefan . The base point free theorem and the Fujita conjecture. School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), 215--248, ICTP Lect. Notes, 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001. Helmke, Stefan . On global generation of adjoint linear systems. Math. Ann. 313 (1999), no. 4, 635--652. Helmke, Stefan . On Fujita's conjecture. Duke Math. J. 88 (1997), no. 2, 201--216. 6 added links; added 26 characters in body; deleted 14 characters in body In the general case when$n=\dim X$is arbitrary Kollár proves an analogue for minimal varieties of general type, although the bound is much worse than in the surface case. In the situation at hand it implies that$\vert ((n+3)!)K_X\vert$is basepoint-free (actually one can write$(n+2)(n+2)!$instead of$(n+3)!$, but this is shorter). See: MR1233485 (94f:14004) 14C20 (14J10) Kollár, János Effective base point freeness. Math. Ann. 296 (1993), no. 4, 595–605. A closely related problem is Fujita's conjecture, which states that if$L$is an ample divisor on$X$, then$|K_X+mL|$is basepoint-free if$m\geq \dim X+1$and very ample if$m\geq \dim X +2$. There are many related results, but the conjecture is still open. Angehrn and Siu prove that if$L$is ample, then$|mL+K_X|$is base point free if$m\geq 1/2(n^2+n+2)$. MR1358978 (97b:32036) Angehrn, Urban(1-HRV)Urban; Siu, Yum Tong(1-HRV) Tong Effective freeness and point separation for adjoint bundles. Invent. Math. 122 (1995), no. 2, 291–308. 32J25 (14C20 32L10) Helmke also has some related results: MR1919459 (2003h:14013) Helmke, Stefan . The base point free theorem and the Fujita conjecture. School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), 215--248, ICTP Lect. Notes, 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001. MR1686947 (2000f:14008) Helmke, Stefan . On global generation of adjoint linear systems. Math. Ann. 313 (1999), no. 4, 635--652. MR1455517 (99e:14003) Helmke, Stefan . On Fujita's conjecture. Duke Math. J. 88 (1997), no. 2, 201--216. 5 added 517 characters in body In the general case when$n=\dim X$is arbitrary Kollár proves an analogue for minimal varieties of general type, although the bound is much worse than in the surface case. In the situation at hand it implies that$\vert ((n+3)!)K_X\vert$is basepoint-free. See: MR1233485 (94f:14004) 14C20 (14J10) Kollár, János Effective base point freeness. Math. Ann. 296 (1993), no. 4, 595–605. A closely related problem is Fujita's conjecture, which states that if$L$is an ample divisor on$X$, then$|K_X+mL|$is basepoint-free if$m\geq \dim X+1$and very ample if$m\geq \dim X +2$. There are many related results, but the conjecture is still open. Angehrn and Siu prove that if$L$is ample, then$|mL+K_X|$is base point free if$m\geq 1/2(n^2+n+2)\$.

MR1358978 (97b:32036) Angehrn, Urban(1-HRV); Siu, Yum Tong(1-HRV) Effective freeness and point separation for adjoint bundles. Invent. Math. 122 (1995), no. 2, 291–308. 32J25 (14C20 32L10)

Helmke also has some related results:

MR1919459 (2003h:14013) Helmke, Stefan . The base point free theorem and the Fujita conjecture. School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), 215--248, ICTP Lect. Notes, 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001.

MR1686947 (2000f:14008) Helmke, Stefan . On global generation of adjoint linear systems. Math. Ann. 313 (1999), no. 4, 635--652.

MR1455517 (99e:14003) Helmke, Stefan . On Fujita's conjecture. Duke Math. J. 88 (1997), no. 2, 201--216.