- Let
`k`be an algebraically closed field. Classify the affine plane curves of degree two (conic curves) over`k`up to isomorphism. - Show that any two curves over an algebraically closed
field are homeomorphic.
### Exercises on Hilbert's Theorems

- Show that both forms of the Nullstellensatz fail over fields that are not algebraically closed.
- Let
`J= < x`. Identify^{2}+ y^{2}+ z^{2}, xy + xz + yz>`Z(J)`and`I(Z(J))`. - Let
`V = Z(xy = zw) Ì A`. Prove that^{4}`A(V)`is not a UFD. - Let
`Y Ì A`be the set^{3}`{ (t, t`. Find^{2}, t^{3}) : t Î k}`I(Y)`. Prove that`A(Y)`is isomorphic to a polynomial ring in one variable. - Let
`I`and`J`be ideals of a ring`R`with`I Ì Rad(J)`. Show that if`I`is finitely generated, then there exists a positive integer`n`such that`I`.^{n}Ì J### Exercises on Morphisms

- Let
`k`be a field of positive characteristic`p`. Consider the morphism`f : A`given by^{1}® A^{1}`f(t) = t`. Show that^{p}`f`is a bijective morphism which is not an isomorphism. - Let
`k`be an algebraically closed field, and let`C`be the algebraic plane curve over`k`defined by`y`. Let^{2}= x^{3}`f : A`be defined by^{1}® C`f(t) = (t`. Show that^{2}, t^{3})`f`is a bijective morphism which is not an isomorphism. - Let
`f = (f`be a morphism, and let_{1}, ¼, f_{n}): A^{n}® A^{n}`J = det[¶f`be the Jacobian polynomial of_{i}/ ¶ x_{j}]`f`. Prove that if`f`is an isomorphism, then`J`is a nonzero constant polynomial. (Note: the converse to this assertion is still an unsolved problem.)### Exercises on Irreducibility

- Let
`Y=Z(J)`be an affine algebraic set. Show that if`J`can be generated by`r`elements, then every irreducible component of`Y`has dimension at least equal to`n - r`. -
Let
`I`be the ideal generated by`<xz - y`in the ring^{2}, yw - z^{2}>`k[x, y, z, w]`. Show that the intersection of the zero set`Z(I)`with the affine space where`w ¹ 0`consists of two irreducible pieces. Use this to show that`I`is the intersection of the ideals`<y,z>`and`I`. Finally, show that the quotient_{0}= <xz - y^{2}, yw - z^{2}, xw - yz>`k[x,y,z,w]/I`is isomorphic to a polynomial ring in two variables._{0} - Let
`J=<xy, xz, yz> Î k[x, y, z]`. Find the irreducible components of`Z(J) Ì A`. Is^{3}`J`radical? prove that`J`cannot be generated by two elements. Now let`J' = < xy, (x-y)z >`, and find the radical of`J'`. - Prove that the product of two irreducible algebraic sets is again irreducible. (Remember that the product in this category does not have the product topology.)
- Find the (three) irreducible components in
`A`of^{3}`<yz=x`.^{2}, x=xz> - Find the irreducible components of
`Z(x`in^{2}= y^{4}, x^{3}= y^{4}-x^{2}y^{2}+xy^{2})`A`.^{3}### Exercises on Rational Functions

- Let
`f`be the map given by_{n}: A^{1}® A^{2}`f`. If_{n}(x) = (x^{2}, x^{n})`n`is odd, show that the image of`f`is isomorphic to_{n}`A`and^{1}`f`is 2-to-1 outside 0. If_{n}`n`is odd, show that`f`is bijective, and provide a rational inverse._{n} - Show that the map
`f : (x:y:z) |® (yz:xz:xy)`is a rational map from`P`to itself. Find the largest open set on which^{2}`f`is defined. Show that`f`is an isomorphism from this set to its image, and find a rational inverse. - Let
`f = Y/X`be a rational function on the affine plane curve`Y`. Find the poles of^{2}= X^{3}+ X^{2}`f`and of`f`.^{2}### Exercises on Birationality

- Let
`k`be an algebraically closed field of characteristic different from`2`. Let`l Î k`with`l ¹ 0, 1`. Show that the affine plane curve defined by

is not rational.`y`^{2}= x(x-1)(x-l) - Prove that the affine plane curve
`y`is rational.^{3}= x^{4}+ x^{3} - Show that the affine plane curve
`Y`defined by`y=x`is isomorphic to^{2}`A`. Show that the affine plane curve^{1}`Z`defined by`xy=1`is birationally isomorphic, but not isomorphic, to`A`. Finally show that every irreducible affine plane curve of degree two is isomorphic either to^{1}`Y`or to`Z`.### Exercises on Projective Space

- Prove that the algebraic sets are the closed subsets of a
topology on
`P`.^{n} - Show that the radical of a homogeneous ideal is homogeneous.
- Prove the "homogeneous Nullstellensatz", which says
that if
`J Ì S = k[x`is a homogeneous ideal and if_{0}, ¼, x_{n}]`f Î S`is a hommogeneous polynomial of positive degree such that`f Î I(Z(J))`, then`f`for some positive integer^{n}Î J`n`. - Let
`k`be an algebraically closed field. Classify the projective plane curves of degree two (conic curves) over`k`up to isomorphism. - A hypersurface in
`P`defined be a linear polynomial is called a hyperplane. A subvariety of^{n}`P`whose ideal is generated by linear polynomials is called a linear variety. Show that a subvariety is linear if and only if it is an intersection of hyperplanes.^{n} - Prove that any two curves in the projective plane have nonempty intersection. In particular, any two distinct lines meet in eactly one point.
- d-uple (iso to image)
- Segre
- quadrics
### Exercises on Sheaves

- Describe the sheaf associated to the constant presheaf
which assigns to all open sets
`U`the same abelian group`A`. - Prove that the exactness of a sequence of morphisms of sheaves can be tested on the stalks.
- Let
`Y`be the topological space with one point. Describe all sheaves of abelian groups on`Y`. Give an example of a ring`R`for which`Y`is homeomorphic to`Spec(R)`. - Let
`Y`be the set with two points, made into a topological space in which all four subsets are open. Describe all sheaves of abelian groups on`Y`. Give an example of a ring`R`for which`Y`is homeomorphic to`Spec(R)`. - Let
`Y`be the set with two points`{0, 1}`, made into a topological space in which only the three subsets`Æ`,`{0}`, and`{0, 1}`are open. Describe all sheaves of abelian groups on`Y`. Give an example of a ring`R`for which`Y`is homeomorphic to`Spec(R)`. - Find the stalks of the sheaves in the previous three problems.
- Prove that a sheaf on a topological space
`X`is completely determined by its sections on a basis for the topology on`X`. - Suppose
`{U`is an open cover of_{i}}`X`, that`F`are sheaves on_{i}`U`, and that_{i}`f`are isomorphisms satisfying the compatibility condition_{ij}: F_{i}½_{UiÇUj}® F_{j}½_{UiÇUj}`f`on triple intersections. Prove that there is a unique sheaf_{jk}f_{ij}= f_{ik}`F`on`X`whose restriction to`U`is isomorphic to_{i}`F`._{i}### Exercises on Spectra

- Let
`R`be the coordinate ring of an affine algebraic variety`X`over an algebraically closed field`k`, and let`P`be the maximal ideal corresponding to a point`x Î X`. Prove that for all`f Î R`, the value`f(P) Î R/P`is just the usual value`f(x) Î k`. - Prove that the Zariski topology on
`Spec(R)`really is a topology. - Describe the topological space
`Spec(C[X])`. - Describe the topological space
`Spec(R[X])`. - Describe the topological space
`Spec(k[X]`._{(x)}) - Describe the topological space
`Spec(k[[X]])`.### Exercises on Affine Schemes

- Let
`X=Spec(R)`and let`P Î X`be a point. Show that there is a one-to-one correspondence between the set of prime ideals in`O`and the set of irreducible closed subsets of_{X,P}`X`that contain`P`.### Exercises on General Schemes

- Show that every scheme is a
`T`topological space._{0}### Exercises on the Functor of Points

### Exercises on Dimension

- Show that every noetherian topological space is quasicompact. Show that every Hausdorff noetherian topological space is a finite set with the discrete topology.
- If
`k`is a field, prove that`P`is a noetherian topological space. Find its dimension.^{n}_{k} - Show that every PID has dimension one.
- Let
`R`be a UFD and let`P=<t>`be a principal, proper, prime ideal. Show that there are no prime ideals properly conatined between`P`and`0`. - Let
`Y`be a linear subvariety of`P`over a field^{n}_{k}`k`. Show that if`dim(Y) = r`, then`I(Y)`is minimally generated by`n - r`linear polynomials. - Let
`Y`and`Z`be linear subvarieties of`P`of dimensions^{n}_{k}`r`and`s`. Show that if`r + s - n ³ 0`, then`Y Ç Z ¹ Æ`. Moreover, if`Y Ç Z ¹ Æ`, then`Y Ç Z`is a linear subvariety of dimension`r + s - n`. - Let
`A`be a noetherian ring of dimension zero. Prove that`A`is artinian. If, an addition,`A`is finitely generated as an algebra over a field`k`, then show that`A`is a finite-dimensional`k`-vector space. - Let
`Y Ì A`be the set^{3}`{ (t`. Prove that^{3}, t^{4}, t^{5}) : t Î k}`I(Y)`is a prime ideal of height 2 that cannot be generated by two elements.### Exercises on Local Properties

- Locate the singular points on the affine plane curves
`y`^{2}= x^{4}+ y^{4}`xy = x`^{6}+ y^{6}`x`^{3}=y^{2}+ x^{4}+ y^{4}`(x`^{2}+ y^{2})^{2}= y^{3}- 3x^{2}y`(x`^{2}+y^{2})^{3}= 4x^{2}y^{2} - Let
`X`be an affine plane curve of degree`n`containing a multiple point`P`of multiplicity`n`. Show that`X`consists of a collection of`n`lines through`P`. - Describe the singular points on the surfaces in
`A`defined by:^{3}`xy=z`^{2}`xy`^{2}=z^{2}`x`^{2}+ y^{2}= z^{2}`xy + x`^{3}+y^{3}= 0 - Suppose
`R`is an algebra over a field`k`that is finitely generated as a`k`-vector space. Prove that`R`is isomorphic to a direct product of artinian local rings. - Let
`R`be a discrete valuation ring containing a field`k`that is isomorphic to its residue field`R/M`. Show that`v(z) = dim`for all_{k}(R/<z`gt``z Î R`.### Exercises on Curves

- Find (birationally isomorphic) nonsingular projective
models of each of the curves in exercise (1) of the section on
*Local Properties*. - Every conic in the projective plane is normal.
- The quadric surfaces
`xy=zw`and`xy=z`in^{2}`P`are normal.^{3} - Show that the cuspidal cubic curve
`y`is not normal.^{2}=x^{3} - Show that an irreducible cubic curve has at most one singular point.

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Kevin R. CoombesDepartment of Biomedical Informatics

The Ohio State University

Columbus, Ohio 43210