# Basics in geometry of curves

Given a smooth curve \(C\) over a field \(k\), the canonical bundle \(K\) is just the bundle of \(1\)-forms, namely \(\Omega_{C/k}^1\). Moreover if \(C/k\) is proper, then for any divisor \(D\) we have Riemann-Roch \[ l(D)-l(K-D)=deg(D)-g+1 \] where \(l(\bullet)\) stands for \(\dim\Gamma(C,\mathcal{O}(\bullet))\), and \(g\) is the genus. In the case of varieties over \(\mathbb{C}\), the (arithmetic) genus is just \(h^1(C,\mathcal{O}_C)\). In fact, according to Serre duality, \(l(K-D)=h^0(\Omega_{C/k}^1\otimes\mathcal{O}(D)^\vee)=h^1(\mathcal{O}(D))\). Therefore the left side is just \(\chi(\mathcal{O}(D))\). The generalized form, Grothendieck-Riemann-Roch formula, states that for a vector bundle \(E/C\) of rank \(n\) and degree \(d\) (the degree of the first Chern class \(c_1(E)\in CH_0(C)\)) we have \[ \chi(C,E)=ch(Rf_*E)=f_*(ch(E)\cdot td(T_{C/k}))=d+n(1-g) \] where \(f:C\rightarrow k\) is the structural morphism.

# Normalization of a curve

It should hold for a curve that the difference \(p_a(C)-p_g(C)\) is equal to the number of points of singular locus of \(C\), which shall vanish after a normalization namely \[ \tilde{C}\xrightarrow{f} C \] and the geometric genus \(p_g(\bullet)\) is birationally invariant.

We can say even more about the normalization map. In fact, \(f\) is uniquely determined by the properties of being finite, birational and \(\tilde{C}\) being normal. If \(C\) is the union of irreducible components \(C_1\cdots C_n\), then \(\tilde{C}\) is the disjoint union of \(\tilde{C}_i\)s (this can be deduced easily from the determining properties above).

# Calculatin genus: Hirzebruch’s formula

# Hilbert scheme and quot scheme

This section is a quick exposure to the theory of Hilbert scheme and Quot scheme. Utilizing hilbert scheme, the ultimate goal is to construct and study the mapping space from a projective scheme to a quasi-projective scheme.

## Quot functor \(\mathfrak{Q}uot_{E/X/S}\) and Hilbert functor \(\mathfrak{H}ilb_{X/S}\)

Given a scheme \(X/S\) of finite
type and a coherent (to expect a good definition of coherence we may
assume \(S\) is at least locally
Noetherian) sheaf \(E\) on \(X\), there is a quot functor defined as
below. For a morphism \(T\rightarrow
S\) of finite type, a **quotient of \(E\) parametrized by \(T\)** is a pair \((\mathcal{F},q)\) satisfying

- \(\mathcal{F}\) is a (quasi-)coherent sheaf on \(X_T\) with schematic support proper over \(T\), and itself flat over \(T\);
- \(q:E_T\rightarrow\mathcal{F}\) is surjective where \(E_T\) is the pullback of \(E\) along \(T/S\).

Two such pairs \((\mathcal{F},q)\) and \((\mathcal{F}',q')\) are considered isomprphic if they share the same kernel in \(E_T\). Since flatness and properness are stable under pullback, such a class parametrized by \(T/S\) can be pullback to \(T'/S\) along \(T'\rightarrow T\). Hence we get a functor \(\mathfrak{Q}uot_{E/X/S}\) sending \(T/S\) to the class of quotents of \(E\) parametrized by \(T\), identifying the isomorphic objects. This is called the quotient functor. The hilbert functor \(\mathfrak{H}ilb_{X/S}\) is defined to be \(\mathfrak{Q}uot_{\mathcal{O}_X/X/S}\).

## Ampleness and positivity

## Hilbert polynomial and the stratification

Given a projective scheme over the field, namely \(X/k\), and a line bundle \(\mathcal{L}\in Pic(X)\), we may associate to every coherent sheaf \(\mathcal{F}\in Coh(X)\) a function, which goes as \[ \Phi^\mathcal{L}(m)=\sum_{i=0}^\infty (-1)^i\dim_kH^i(X,\mathcal{F}\otimes\mathcal{L}^{\otimes m}). \] Here we do not assume any property of \(\mathcal{L}\), however we may see later that it’s of most convenience to choose an ample bundle. Since \(X\) is projective, the right derivation of the projection \(f:X\rightarrow k\) has a finite cohomological dimension, hence the sum make sense, and it’s a classical result that \(\Phi^\mathcal{L}\) is a numerical polynomial, which is then named as the Hilbert polynomial (with respect to \(\mathcal{L}\)). The first property of this construction is quite straightforward: the polynomial should be invariant under an extension of the base field.

Let \(\mathcal{M}=\mathcal{F}\otimes\mathcal{L}^{\otimes m}\in Coh(X)\) and \(\mathcal{M}'=\mathcal{M}\otimes_k k'\), we know that \([Rs_*Rf'_*\mathcal{M}']=[Rf_*Rs'_*\mathcal{M}']=[Rf_*Rs'_*L(s')^*\mathcal{M}]=[k':k][Rf_*\mathcal{M}]\), and furthermore \([k':k]dim_{k'}[Rf_*'\mathcal{M}']=dim_{k}[Rs_*Rf'_*\mathcal{M}']\), which gives the expected result.

Assuming \(\mathcal{L}\) being very ample, the second property, which is more involving, goes that for a projective scheme \(X\) over a locally Noetherian base \(S\) and a coherent sheaf \(\mathcal{F}\in Coh(X)\) which is flat over \(S\), the Hilbert polynomials over fibres of \(S\), viewed as a map sending the spectra of \(S\) to polynomials, should be locally constant on \(S\). To prove this, we may further assume that \(S\) is local, and \(X\) is projective over \(S\).

### Lemma: Serre’s vanishing theorem

Let \(S=Spec(A)\), with \(A\) noetherian, let \(X\) be a projective scheme over \(S\) and let \(\mathcal{L}\) be an ample line bundle on \(X\). Then for all \(\mathcal{F}\in Coh(X)\), there exists an integer \(n_0\) , such that \(H^q(X,\mathcal{F}(n))=0\) for all \(q>0\) and \(n\geq n_0\) , where \(\mathcal{F}(n)=\mathcal{F}\otimes\mathcal{L}^{\otimes n}\).

### Lemma: \(\mathcal{F}\in Coh(X)\) is flat over \(S\) iff \(H^0(X,\mathcal{F}(d))\) is finite free for \(d\) large enough.

Let there be a affine covering \(\mathcal{U}=(U_i)\) of \(X\), then the Cech resolution (note that projective morphisms are automatically sperarated) \[ 0\rightarrow H^0(X,\mathcal{F}(d))\rightarrow C^0(\mathcal{U},\mathcal{F}(d))\rightarrow\cdots\rightarrow C^m(\mathcal{U},\mathcal{F}(d))\rightarrow 0 \] is exact for \(d\) large enough. Since all the \(C^i(\bullet)\)s are flat and finitely generated, so is \(H^0(X,\mathcal{F}(d))\), which is then free.

On the other hand, ()

## Dimension \(0\) case: \(Hilb_1=?\)

Say \(X/k\) is a projective variety, with a very ample line bundle \(\mathcal{O}(1)\) which defines the embedding of \(X\) in \(\mathbb{P}_k^n\) for some \(n\).

Fix a closed subscheme \(Z\) in \(X\) with structure sheaf \(\mathcal{O}_Z\), then in this case we have \(Rf_*(\mathcal{O}_Z(m))=R(f_0)_*\circ Ri_*(\mathcal{O}(m))\). Note that \(i\) is affine so the derived pushforward can be directly caculated in the projective space \(\mathbb{P}_k^n\), where \(Z\) viewed as a closed subscheme. Given the fact that \(Z\) admits a constant Hilbert polynomial, on knows immediately that \(Z\) is of dimension \(0\) in \(X\), and of degree equal to that polynomial. Despite of the fact that degree of a closed immersion depends on the selection of an ample bundle and is hence not well defined, degree of a zero-dimensional subscheme is well defined indeed. Assume furthermore that \(Z\) is of degree \(1\), then since \(h^0(Z,\mathcal{O}_Z)=1\), the only possible for \(Z\) is to be a reduced rational point itself. Hence \(Hilb_1(X/k)\) is just \(X/k\).

## Dimension \(0\) case: \(Hilb_m=?\)

# Mapping space, ready for doing deformation

## Tangent space

# Specialization of cycles

# Chow variety, the Hilbert-Chow morphism

Different from Hilbert scheme, Chow variety is the moduli space of cycles instead of quotients. Namely, the variety \(Gr(l,d,n)\) over a field \(k\) parametrizes all the effective cycles of dimension \(l\) and degree \(d\) in the space \(\mathbb{P}^{n-1}\).

To illustrate the case, note that \(\mathbb{A}^2/(x,y^2)\) and \(\mathbb{A}^2/(x+y,x^2+y^2)\) are two pairs of duplicated points having distinct tangent spaces. However, they determine the same cycle in \(\mathbb{A}^2\), which suggests that there should be a “forgetful” or “cycle” morphism from Hilbert scheme to the corresponding Chow variety. This is named as the Hilbert-Chow morphism.

## Examples of Chow varieties

The simplest example is the case where \(d=1\), the variety \(Gr(l,1,n)\) is just the Grassmannian \(Gr(l,n)\), which parametrizes hypersurfaces of dimension \(l\) embedded in \(\mathbb{P}^{n-1}\).