Definitions
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What is a prime divisor?
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What is a Weil divisor?
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What is a Cartier divisor?
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What is an effective divisor?
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What is a reduced divisor?
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What is a principal divisor?
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What is the divisor class group?
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What does it mean for divisors to be linearly equivalent?
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What does it mean for two divisors to be numerically equivalent?
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What is the Picard group?
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What is the pullback of a divisor?
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What is the divisor associated to a section of a bundle?
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For \(D\) a prime divisor and \(p\in \mathop{\mathrm{supp}}D\), what is the multiplicity of \(p\) in \(D\)?
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What is a SNC divisor?
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For \(D\) a divisor, what is \({\mathcal{O}}_X(D)\)?
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What is a polarization?
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What is an ample divisor?
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What is a very ample divisor?
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What is a nef divisor?
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What is a big divisor?
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What is a variety of general type?
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What is the ramification divisor of a surjective morphism \(f:X\to Y\) of smooth projective curves?
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What is the ramification index?
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What is the branch locus?
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What is a linear system?
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What is a complete linear system?
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What is the complete linear system associated to a section?
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What is the base locus of a linear system?
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What does it mean for a linear system to be base point free?
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What is the Riemann-Roch space?
Problems
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Show that the hyperplane sections of a projective variety \(X\) form a base point free linear system of effective divisors on \(X\).
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Show that if \(X\) is normal, then a generic element of this system is a smooth and reduced divisor.