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Ruadhaí Dervan
Ruadhaí Dervan
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In general there may not be smooth anticanonical divisors. Of course if you take divisors in the linear system $|-mK_X|$ for $\gg 1$ there will be lots of smooth divisors, by Bertini's theorem (precisely when $|-mK_X|$ is basepoint free). But if you restrict to $m=1$ this is not always the case. Precisely, if |-K_X| is basepoint free there will be lots of smooth anticanonical divisors.

For an example of a Fano manifold without a smooth anticanonical divisor, see the work of Taro Sano: http://arxiv.org/abs/1302.0705 Example 2.9. Apparently the first such examples are due to Hoering-Voisin: http://arxiv.org/abs/1009.2853 Example 2.12.

In general there may not be smooth anticanonical divisors. Of course if you take divisors in the linear system $|-mK_X|$ there will be lots of smooth divisors, by Bertini's theorem. But if you restrict to $m=1$ this is not always the case. Precisely, if |-K_X| is basepoint free there will be lots of smooth anticanonical divisors.

For an example of a Fano manifold without a smooth anticanonical divisor, see the work of Taro Sano: http://arxiv.org/abs/1302.0705 Example 2.9. Apparently the first such examples are due to Hoering-Voisin: http://arxiv.org/abs/1009.2853 Example 2.12.

In general there may not be smooth anticanonical divisors. Of course if you take divisors in the linear system $|-mK_X|$ for $\gg 1$ there will be lots of smooth divisors, by Bertini's theorem (precisely when $|-mK_X|$ is basepoint free). But if you restrict to $m=1$ this is not always the case.

For an example of a Fano manifold without a smooth anticanonical divisor, see the work of Taro Sano: http://arxiv.org/abs/1302.0705 Example 2.9. Apparently the first such examples are due to Hoering-Voisin: http://arxiv.org/abs/1009.2853 Example 2.12.

Source Link
Ruadhaí Dervan
Ruadhaí Dervan
  • 1.4k
  • 10
  • 18

In general there may not be smooth anticanonical divisors. Of course if you take divisors in the linear system $|-mK_X|$ there will be lots of smooth divisors, by Bertini's theorem. But if you restrict to $m=1$ this is not always the case. Precisely, if |-K_X| is basepoint free there will be lots of smooth anticanonical divisors.

For an example of a Fano manifold without a smooth anticanonical divisor, see the work of Taro Sano: http://arxiv.org/abs/1302.0705 Example 2.9. Apparently the first such examples are due to Hoering-Voisin: http://arxiv.org/abs/1009.2853 Example 2.12.