57 problems.
General Theory
- Nonempty Zariski-open subsets of an affine variety are dense
- The Zariski topology on a curve is the cofinite topology
- The complement of a hypersurface is affine, and its coordinate ring
- When the coordinate ring \(k[X]\) is an integral domain
- Smoothness against normality
- Unique decomposition of an affine algebraic set into irreducible components
- Normal varieties are smooth in codimension one
- Irreducible closed subvarieties of \(\AA^2\)
- A projective variety is irreducible iff its ideal is homogeneous prime
- The ideal of a subset of \(\AA^n\) is radical, and maximal exactly at points
- The polynomial ring on affine space is Noetherian
- The Zariski topology is never Hausdorff except on a point
- \(\operatorname{Specm} k[x]\) carries the cofinite topology
- The coordinate ring of a product of affine varieties is a tensor product
- Self-morphisms of the affine line and of the punctured line are finite
- Morphisms from the affine line to the punctured line are constant
- Quasi-finite morphisms do not raise dimension
- Equidimensional finite morphisms are closed and surjective
- Existence of a non-surjective finite morphism
- Dimension of the fibers of a dominant morphism of affine varieties
- Components of the zero locus of a nonconstant regular function have codimension one
- The tangent space has dimension equal to \(\dim X\) exactly at smooth points
- Smooth points are exactly the points where \(X\) is locally a submanifold
- The tangent space as the dual of \(\mfm_p/\mfm_p^2\)
- The singular locus is a proper Zariski closed subset
- Normal affine curves are smooth
- Zariski closed subsets of \(\PP^n\) are compact in the classical topology
- Irreducibility of projective varieties
- The singular locus of a normal projective variety has codimension at least two
- Normal projective curves are smooth
- Projection along a projective factor is proper and closed
- Morphisms of complex projective varieties: properness, images, and birationality
- Singular points and tangent lines of a plane projective curve via the gradient
- Whether isomorphic varieties have isomorphic affine cones
- The automorphism group of \(\PP^n\) is \(\PGL_{n+1}\)
- Birational morphisms between smooth projective curves are isomorphisms
- The projective closure as the variety together with its boundary at infinity
- The cohomology ring of projective space
- Smooth plane cubics are elliptic curves
- Every genus zero smooth projective curve over \(\CC\) is a plane conic
- Maps of finite degree between genus one curves are unramified
- An everywhere regular differential \(n\dash\)form on \(\AA^n\)
- The canonical bundle of projective space
- Global regular functions on a connected complete variety are constant
Examples
- \(\GL_n(\CC)\) is an affine variety
- The punctured affine plane is not affine
- The affine cubic \(V(x(xy-1))\) has two irreducible components
- The hyperbola \(V(xy-1)\) is not isomorphic to \(\AA^1\)
- The map \((x,y)\mapsto (x, xy)\) of the affine plane and its fibers
- The cuspidal cubic, its unique singular point, and its normalization
- The nodal cubic, its two smooth branches, and Zariski’s main theorem
- The cyclic quotient singularity \(\AA^2/\mu_n\)
- The Segre image of \(\PP^1\cross \PP^1\) is a smooth quadric surface
- Nonsingularity of the Weierstrass cubic and the discriminant
- The class group of the quadric cone \(V(xy-z^2)\)
- The class group of \(V(y^2-x(x^2-1))\)
- The Picard group of projective space