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    Miscellaneous algebraic geometry problems

    SRC-AGX-MISC-AG-PROBLEMS

    Area
    Algebraic Geometry
    Status
    draft
    Contents

    11 problems.

    Curves and Rationality

    1. \(\PP^n\cross \PP^m\) is a rational variety
    2. Maps of finite degree between genus one curves are unramified
    3. Every genus zero smooth projective curve over \(\CC\) is a plane conic
    4. \(\OO_Y \to \pi_* \OO_X\) is an isomorphism exactly when \(Y\) is integrally closed in \(K(X)\)
    5. A family of genus-zero curves over a curve is a \(\PP^1\)-bundle

    Sheaves of Modules

    1. Nonzero maps of invertible sheaves on an integral scheme are injective

    Irreducibility

    1. A space is irreducible exactly when its nonempty open subsets are connected
    2. The cuspidal cubic is irreducible
    3. Two irreducible subvarieties of \(\PP^3\) with reducible intersection
    4. Irreducible components of two algebraic sets

    Calculations

    1. Oral calculations on projective space, sections and divisors

    Browse these problems in source order