5_3x

V.3: Monoidal Transformations

V.3.1. #to_work

Let \(X\) be a nonsingular projective variety of any dimension, let \(Y\) be a nonsingular subvariety, and let \(\pi: \tilde{X} \rightarrow X\) be obtained by blowing up \(Y\). Show that \(p_a(\tilde{X})=\) \(p_a(X)\)

V.3.2. #to_work

Let \(C\) and \(D\) be curves on a surface \(X\), meeting at a point \(P\). Let \(\pi: \tilde{X} \rightarrow X\) be the monoidal transformation with center \(P\).

Show that \begin{align*} \tilde{C} \cdot \tilde{D}=C . D-\mu_P(C) \cdot \mu_P(D) .\end{align*} Conclude that \(C . D=\sum \mu_P(C) \cdot \mu_P(D)\), where the sum is taken over all intersection points of \(C\) and \(D\), including infinitely near intersection points.

V.3.3. #to_work

Let \(\pi: \tilde{X} \rightarrow X\) be a monoidal transformation, and let \(D\) be a very ample divisor on \(X\). Show that \(2 \pi^* D-E\) is ample on \(\tilde{X}\). ¹

V.3.4. Multiplicity of a Local Ring. #to_work

Let \(A\) be a noetherian local ring with maximal ideal \({\mathfrak{m}}\). For any \(l>0\), let \(\psi(l)=\) length \(\left(A / {\mathfrak{m}}^l\right)\). We call \(\psi\) the Hilbert-Samuel function of \(A\).

• Show that there is a polynomial \(P_A(z) \in \mathbf{Q}[z]\) such that \(P_A(l)=\psi(l)\) for all \(l \gg 0\). This is the Hilbert-Samuel polynomial of \(A\). ^{2 3}

• Show that \(\operatorname{deg} P_A=\operatorname{dim} A\).

• Let \(n=\operatorname{dim} A\). Then we define the multiplicity of \(A\), denoted \(\mu(A)\), to be \((n !)\). (leading coefficient of \(P_A\) ). If \(P\) is a point on a noetherian scheme \(X\), we define the multiplicity of \(P\) on \(X, \mu_P(X)\), to be \(\mu\left(\mathcal{O}_{P, X}\right)\).

• Show that for a point \(P\) on a curve \(C\) on a surface \(X\), this definition of \(\mu_P(C)\) coincides with the one in the text just before (3.5.2).

• If \(Y\) is a variety of degree \(d\) in \(\mathbf{P}^n\), show that the vertex of the cone over \(Y\) is a point of multiplicity \(d\).

V.3.5. #to_work

Let \(a_1, \ldots, a_r, r \geqslant 5\), be distinct elements of \(k\), and let \(C\) be the curve in \(\mathbf{P}^2\) given by the (affine) equation \(y^2=\prod_{i=1}^r\left(x-a_i\right)\). Show that the point \(P\) at infinity on the \(y\)-axis is a singular point. Compute \(\delta_P\) and \(g(\tilde{Y})\), where \(\tilde{Y}\) is the normalization of \(Y\). Show in this way that one obtains hyperelliptic curves of every genus \(g \geqslant 2\).

V.3.6. #to_work

Show that analytically isomorphic curve singularities (I, 5.6.1) are equivalent in the sense of (3.9.4), but not conversely.

V.3.7. #to_work

For each of the following singularities at \((0,0)\) in the plane, give an embedded resolution, compute \(\delta_P\), and decide which ones are equivalent.

• \(x^3+y^5=0\).

• \(x^3+x^4+y^5=0\).

• \(x^3+y^4+y^5=0\).

• \(x^3+y^5+y^6=0\).

• \(x^3+x y^3+y^5=0\).

V.3.8. #to_work

Show that the following two singularities have the same multiplicity, and the same configuration of infinitely near singular points with the same multiplicities, hence the same \(\delta_P\), but are not equivalent.

• \(x^4-x y^4=0\).

• \(x^4-x^2 y^3-x^2 y^5+y^8=0\).