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Bertini's theorem (see Griffiths and Harris, Principles of Algebraic Geometry) states that:The generic element of a linear system is smooth away from the base locus of the system.

My questions are:

1)What is "generic element of a linear system"?

2)what does it mean " smooth away from the base locus of the system"?

Thank you!

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    $\begingroup$ A generic element of a linear system is simply an element of the linear system which is outside a proper closed subset, as "generic" always means in geometry. Something is "smooth away from the base locus of the system" if it is smooth at its points which are not in the base locus of the system (what else could it mean?!) $\endgroup$ – Mariano Suárez-Álvarez Mar 21 '18 at 0:08
  • $\begingroup$ And what does a generic element (a divisor in the linear system that does not belong to the base locus) be smooth? $\endgroup$ – Manoel Mar 21 '18 at 2:36
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    $\begingroup$ Hm. I do not understand what you are asking. $\endgroup$ – Mariano Suárez-Álvarez Mar 21 '18 at 2:38
  • $\begingroup$ Sorry. I do not know what means an element $D \not \in \{ $base locus$\}$ to be smooth. $\endgroup$ – Manoel Mar 21 '18 at 2:45
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    $\begingroup$ Each element of a linear system is a divisor, so it is a subvariety. It contains the base locus of the system (which is the set of points which belongs to all the divisors of the system) and the clan is that almost all of them are smooth except possibly at the points which are in the base locus. $\endgroup$ – Mariano Suárez-Álvarez Mar 21 '18 at 3:01

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