Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 15 (Basic Algebraic Geometry)

We work over an algebraically closed field $k$, so the assertion concerns a geometric projective line. Let $V=k^4$ and let $W$ be the vector space of homogeneous cubic forms on $V$. It has dimension $\left(\genfrac{}{}{0ex}{}{6}{3}\right)=20$, so the parameter space of cubic equations up to nonzero scalar is $\mathbb{P}(W)={\mathbb{P}}^{19}$. The Grassmannian $G=\mathrm{Gr}(2,V)$ parametrizes lines in ${\mathbb{P}}^3$. Its standard graph charts are four-dimensional affine spaces, so its dimension is four. The Plücker embedding realizes it as a closed quadric in ${\mathbb{P}}^5$, proving projectivity. The general linear group ${\mathrm{GL}}_4$ acts transitively on two-planes and is an irreducible open subset of the affine space of matrices; its orbit map proves that $G$ is irreducible.

Consider the incidence varietyRestriction of a cubic to a fixed line is surjective onto the four-dimensional space of binary cubic forms: after choosing coordinates with that line given by $x_2=x_3=0$, extend any binary cubic using $x_0,x_1$ alone. Thus its kernel has dimension sixteen. On each Grassmannian chart these restriction maps depend algebraically on the line parameters and have constant rank four. Their kernels form a rank-sixteen vector bundle, and $\mathcal{I}$ is its projective bundle, with fibre ${\mathbb{P}}^{15}$. In particular, $\mathcal{I}$ is irreducible of dimension $4+15=19$. It is closed in the product, since restriction vanishes precisely when four polynomial coefficients vanish.

The projection $\pi :\mathcal{I}\to \mathbb{P}(W)$ is a projective morphism, hence its image is closed. It remains to prove that this image has dimension nineteen. Equal dimensions alone would not suffice: we must control a fibre locally.

TakeIn the open Grassmannian chart around ${\ell }_0$, every line is uniquely a graphRestriction of $F_0$ to that line isThe four monomials are linearly independent over every field. Hence the only line in this chart lying on $F_0$ has $a_0=a_1=b_0=b_1=0$. The fibre ${\pi }^{-1}([F_0])$ therefore has local dimension zero at $({\ell }_0,[F_0])$. We do not need $F_0$ to define a smooth surface, nor do we claim its entire fibre is finite.

The fiber dimension theorem, in its local dimension inequality, givesThus the closed image has dimension at least nineteen. A proper closed subset of the irreducible ${\mathbb{P}}^{19}$ has smaller dimension, so $\pi (\mathcal{I})={\mathbb{P}}^{19}$. Every cubic equation consequently has a line in its zero set. In particular, every nonsingular cubic surface contains a line. The proof in fact establishes existence for arbitrary cubic surfaces over an algebraically closed field; it does not assert a rational line over an arbitrary nonclosed ground field.

Use the graded ring homomorphismwhere source degree $m$ maps to target degree $3m$. The homogeneous ideal of the twisted cubic is $\ker \phi$: a homogeneous polynomial vanishes on the parametrized curve precisely when its substitution vanishes identically. Here $k$ is infinite, being an algebraically closed field, so vanishing at all parameter points implies the polynomial identity.

Let $J$ be generated byTheir substitutions vanish, hence $J\subseteq \ker \phi$. To prove the reverse inclusion as an equality of ideals, we need more than a set-theoretic description of the zero locus.

Modulo $J$, replace $x_1^2$ by $x_0{x}_2$, $x_2^2$ by $x_1{x}_3$, and $x_1{x}_2$ by $x_0{x}_3$. Each replacement strictly decreases the sum of the exponents of $x_1$ and $x_2$, so the reduction terminates. It shows that the degree-$m$ quotient is spanned by the standard monomials of the twisted cubicFor $m\ge 1$ there are $(m+1)+m+m=3m+1$ of these. For $m=0$ the sole monomial is $1$. Their substituted images have $t$-exponents respectively $3b$, $3b+1$ and $3b+2$, so they are exactly the distinct monomialsThese are linearly independent. Consequently any degree-$m$ polynomial in $\ker \phi$, after reduction modulo $J$, has all its reduced coefficients zero. It belongs to $J$. Applying this to each homogeneous component givesThis also proves that the quotient embeds in the domain $k[s,t]$, so the displayed ideal is prime; no radicalization or saturation correction is concealed in the argument. Its image is the third Veronese subring ${⨁}_{m\ge 0}k[s,t{]}_{3m}$, and the calculation gives the Hilbert polynomial $3m+1$, consistent with degree three and genus zero.

The usual field norm $N_{L/K}$ is defined for a finite field extension. The printed hypothesis “algebraic” alone allows an infinite extension, for which this determinant norm is not defined. We therefore assume $[L:K]<\infty$. For an arbitrary algebraic extension the meaningful elementwise statement is instead $N_{K(b)/K}(b)\in A$, and we prove this as well.

First a unique factorization domain is an integrally closed domain. Indeed, write an integral fraction as $u/v$ with relatively prime numerator and denominator. A monic integral equation, after multiplication by $v^r$, shows that $v$ divides $u^r$. Unique factorization forces $v$ to be a unit.

Let $m_b(T)=T^r+a_{r-1}{T}^{r-1}+\cdots +a_0\in K[T]$ be the monic minimal polynomial of $b$. Because $b$ is an integral element over $A$, it satisfies a monic polynomial $f\in A[T]$, and $m_b$ divides $f$ in $K[T]$. Every root of $m_b$ in an algebraic closure is therefore integral over $A$. This assertion includes repeated roots when the extension is inseparable. The coefficients of $m_b$ are elementary symmetric expressions in these roots. The integral elements form a ring: finitely many integral elements generate a finite $A$-module algebra, and the determinant trick shows every element of that algebra is integral. Thus all $a_i$ are integral over $A$. Since $a_i\in K$ and $A$ is integrally closed, $a_i\in A$.

Multiplication by $b$ on $K(b)$, in the basis $1,b,\dots ,b^{r-1}$, has the companion matrix of $m_b$, whose determinant is $(-1{)}^r{a}_0$. If $e=[L:K(b)]$, take a $K(b)$-basis of $L$. Multiplication by $b$ preserves each summand in the resulting direct sum of $e$ copies of $K(b)$, so its determinant over $K$ isIn particular $N_{K(b)/K}(b)=(-1{)}^r{a}_0\in A$. The argument requires no separability; in fact it holds for every integrally closed domain, not just a unique factorization domain.

For the connection with dimension, let $B$ be the coordinate ring of an irreducible affine variety. By Noether normalization there is a finite integral inclusion $A=k[t_1,\dots ,t_d]\subset B$, with $d=\dim B$. Their fraction fields form a finite extension. For any $0\ne b\in B$, its minimal-polynomial constant term $a_0$ is nonzero and lies in $A$. Its equation givesThus a nonzero function on the normalized variety produces a nonzero function on the base that vanishes wherever $b$ vanishes. Up to sign, the field norm is a power of this $a_0$, so it too belongs to $bB\cap A$.

Under the finite morphism to ${\mathbb{A}}^d$, the image of $V_B(b)$ is therefore contained in the proper Zariski-closed set $V_A(a_0)$ of dimension at most $d-1$. Finite morphisms preserve the dimension of a closed subvariety and its image. Applying this to a nonzero element of the ideal of any proper closed subvariety proves proper closed subvarieties have strictly smaller dimension. This is the useful role of the norm argument: nonzero equations upstairs give nonzero equations downstairs under a finite normalization.

Let $X$ be a separated Noetherian scheme over a field $k$ and $\mathcal{F}$ a coherent sheaf. Coherent cohomology studies the groups $H^q(X,\mathcal{F})$: global sections in degree zero and the obstructions to gluing local sections in higher degrees. Coherence means finite generation locally, and on a Noetherian scheme it is preserved by kernels, cokernels and extensions. This makes coherent sheaves suitable for exact-sequence calculations.

For an ordered open cover $\mathcal{U}=(U_i)$, the Čech cochain groups arewith differentialEach double omission occurs twice with opposite signs, so ${\delta }^2=0$. The Čech cohomology of the cover is ${\check{H}}^q=\ker \delta /\mathrm{im}\delta$, and taking the direct limit over refinements defines cover-independent Čech cohomology. In degree zero the sheaf gluing axiom identifies it with $\Gamma (X,\mathcal{F})$.

The comparison with derived-functor sheaf cohomology is particularly effective algebraically. Positive-degree cohomology of a quasi-coherent sheaf on an affine scheme vanishes. A separated scheme has affine intersections of affine opens, so a finite affine cover is acyclic for $\mathcal{F}$. Its Čech cochain complex consequently computes $H^q(X,\mathcal{F})$ directly. For a short exact sequence of sheaves, sections over each affine intersection are exact on quasi-coherent sheaves. The resulting short exact sequence of Čech complexes gives the long exact sequence in sheaf cohomology. In particular, the connecting map measures the failure of a global section of the quotient to lift globally.

Several central projective results make the theory finite and computable. On a projective variety over $k$, every coherent cohomology group is finite-dimensional. The Grothendieck vanishing theorem gives $H^q(X,\mathcal{F})=0$ for $q>\dim X$. If ${\mathcal{O}}_X(1)$ is ample, Serre vanishing gives $H^{q>0}(X,\mathcal{F}(m))=0$ for sufficiently large $m$, and these large twists are globally generated sheaves. On projective space a coherent sheaf has a finite twisting resolution of a coherent sheaf on projective space: finite sums of $\mathcal{O}(a)$ resolve it. The explicit line-bundle cohomology below and the long exact sequences prove finiteness and positive-twist vanishing. For a closed embedding $i:X\hookrightarrow {\mathbb{P}}^n$, $i_{\ast }\mathcal{F}$ is coherent and $H^q(X,\mathcal{F})=H^q({\mathbb{P}}^n,i_{\ast }\mathcal{F})$, reducing these results to projective space. A sufficiently positive twist of a finite presentation by sums of $\mathcal{O}(-a)$ also proves global generation.

The Euler characteristic of a coherent sheafis therefore defined, and the long exact sequence makes it additive. With a fixed projective embedding, $\chi (X,\mathcal{F}(m))$ is its Hilbert polynomial, and for sufficiently large $m$ it equals $h^0(X,\mathcal{F}(m))$. On a smooth projective variety of dimension $r$, Serre duality gives, for a vector bundle $E$,For an arbitrary coherent sheaf the correct counterpart is $H^q(X,\mathcal{F}{)}^{\ast }\cong {\mathrm{Ext}}_X^{r-q}(\mathcal{F},{\omega }_X)$; replacing this Ext group indiscriminately by the cohomology of the ordinary sheaf dual would lose local extension information.

We now calculate the basic examples. For $n\ge 1$, use the standard cover $U_i=D_{+}(x_i)$ of ${\mathbb{P}}^n$. Sections of $\mathcal{O}(m)$ on an intersection are degree-$m$ Laurent monomials in which negative exponents are allowed only for coordinates inverted on that intersection. Decompose the Čech cochain complex by exponent vectors $(a_0,\dots ,a_n)$ with sum $m$. If no exponent is negative, the corresponding simplex complex contributes only in degree zero. If all exponents are negative, the monomial occurs only on the full intersection and contributes only in degree $n$. When the negative-index set is nonempty and proper, inserting any index outside it gives a contracting homotopy, so the monomial complex is acyclic. Hence the cohomology of twisting sheaves on projective space isIn the top group a basis consists of the Laurent monomials with every exponent at most $-1$. Writing $a_i=-1-b_i$ reduces the count to nonnegative solutions of $\sum b_i=-m-n-1$, giving the displayed binomial coefficient. In particular, all intermediate cohomology vanishes, and ${\omega }_{{\mathbb{P}}^n}=\mathcal{O}(-n-1)$ agrees with Serre duality. For ${\mathbb{P}}^0$ separately, every twist is trivial and $H^0=k$.

On the projective line this givesThus positive and negative degrees produce quite different groups, while their alternating difference is a single polynomial. The twisted cubic is isomorphic to ${\mathbb{P}}^1$ and its hyperplane bundle corresponds to ${\mathcal{O}}_{{\mathbb{P}}^1}(3)$. Consequently $h^0(X,{\mathcal{O}}_X(m))=3m+1$ for $m\ge 0$, agreeing with Q2, and $h^1(X,{\mathcal{O}}_X(m))=\max (-3m-1,0)$.

For a smooth plane curve $C$ of degree $d\ge 1$, the ideal-sheaf sequence of a projective hypersurface isThe long exact sequence in sheaf cohomology and the preceding formula give $H^0(C,{\mathcal{O}}_C)=k$ and $H^1(C,{\mathcal{O}}_C)\cong H^2({\mathbb{P}}^2,\mathcal{O}(-d))$. ThereforeFor $d=1,2$ this is zero; for a plane cubic it is one. Twisting the same exact sequence gives $\chi (C,{\mathcal{O}}_C(m))=dm+1-g$. These examples illustrate how Čech cohomology, vanishing, Serre duality and exact sequences convert local equations into global numerical invariants.

Use algebraic Chern classes in the Chow ring of a nonsingular algebraic variety $Y$. For a vector bundle $E$ of rank $r$, the classes $c_i(E)\in {\mathrm{CH}}^i(Y)$ are invariant under bundle isomorphism, have $c_0(E)=1$, vanish for $i>r$, and commute with pullback. Their Total Chern class is $c(E)={\sum }_i{c}_i(E)$. The Whitney sum formula for Chern classes applies to every short exact sequence $0\to E^{\prime }\to E\to E^{\prime \prime }\to 0$:Normalization for a line bundle is $c({\mathcal{O}}_Y(D))=1+[D]$, where $[D]$ is the divisor class. In particular, $c_1(L\otimes M)=c_1(L)+c_1(M)$, $c_1(L^{-1})=-c_1(L)$, and the trivial bundle has total class one. These properties determine the classes: pull back to the iterated projective bundle where $E$ has a filtration by line bundles. The projective bundle formula for Chow groups makes this pullback injective, and Whitney multiplication expresses $c_i(E)$ as the elementary symmetric polynomial of the first classes of the line factors. This also gives $c_i(E^{\ast })=(-1{)}^i{c}_i(E)$.

Let $X$ be the smooth degree-$d$ projective hypersurface and set $h=c_1({\mathcal{O}}_X(1))$. The algebraic cotangent bundle has rank $n-1$. Restriction of the cotangent Euler sequence in homogeneous coordinates remains exact, because its terms and quotient are locally free. Thusand Whitney multiplication givesThe defining equation has degree $d$, so the conormal sheaf is ${\mathcal{O}}_X(-d)$. Smoothness makes its differential map injective with locally free quotient, giving the Conormal exact sequence for Kähler differentialsA second application of the Whitney sum formula for Chern classes yieldsThe inverse is the finite geometric series in the graded Chow ring: terms in codimension greater than $n-1$ vanish. Expanding gives every requested class,For example,when the corresponding ranks and codimensions exist. The first expression agrees with the canonical bundle of a smooth projective hypersurface $\det {\Omega }_X^1={\mathcal{O}}_X(d-n-1)$. For a plane curve, $\mathrm{deg}{c}_1=d(d-3)=2g-2$, matching Q4. For a quartic surface in ${\mathbb{P}}^3$, $c_1=0$ and $c_2=6h^2$, whose degree is $24$. These checks also confirm the sign in the cotangent, rather than tangent, formula.