91Ó°ÊÓ

\(f_{*}\) for Divisors. Let \(f: X \rightarrow Y\) be a finite morphism of curves of degree \(n .\) We define a homomorphism \(f_{*}:\) Div \(X \rightarrow\) Div \(Y\) by \(f_{*}\left(\sum n_{i} P_{i}\right)=\sum n_{i} f\left(P_{i}\right)\) for any divisor \(D=\sum n_{i} P_{i}\) on \(X\) (a) For any locally free sheaf \(\delta\) on \(Y\), of rank \(r,\) we define det \(\mathscr{B}=\wedge^{r} \mathscr{E} \in \operatorname{Pic} Y\) (II, Ex. 6.11 ). In particular, for any invertible sheaf \(\mathscr{M}\) on \(X, f_{*} . / /\) is locally free of rank \(n\) on \(Y\), so we can consider det \(f_{*} \mathscr{H} \in\) Pic \(Y\). Show that for any divisor \(D\) on \(X\) $$\operatorname{det}\left(f_{*} \mathscr{L}(D)\right) \cong\left(\operatorname{det} f_{*} c_{X}\right) \otimes \mathscr{L}\left(f_{*} D\right)$$ Note in particular that \(\operatorname{det}\left(f_{*} \mathscr{L}(D)\right) \neq \mathscr{L}\left(f_{*} D\right)\) in general! \([\text { Hint }:\) First consider an effective divisor \(D,\) apply \(f_{*}\) to the exact sequence \(0 \rightarrow \mathscr{P}(-D) \rightarrow\) \(\left.\varphi_{x} \rightarrow \mathscr{C}_{D} \rightarrow 0, \text { and use (II, Ex. } 6.11 \text { ). }\right]\) (b) Conclude that \(f_{*} D\) depends only on the linear equivalence class of \(D,\) so there is an induced homomorphism \(f_{*}: \operatorname{Pic} X \rightarrow\) Pic \(Y\). Show that \(f_{*} \cdot f^{*}: \operatorname{Pic} Y \rightarrow\) Pic \(Y\) is just multiplication by \(n\) (c) Use duality for a finite flat morphism (III, Ex. 6.10 ) and (III, Ex. 7.2) to show that $$\operatorname{det} f_{*} \Omega_{X} \cong \text { (det } f_{*}(x)^{-1} \otimes \Omega_{Y}^{\otimes n}$$ (d) Now assume that \(f\) is separable, so we have the ramification divisor \(R\). We define the branch divisor \(B\) to be the divisor \(f_{*} R\) on \(Y\). Show that $$\left(\operatorname{det} f_{*} C_{x}\right)^{2} \cong \mathscr{L}(-B)$$

Step by step solution

01

Set the ground

Given a morphism \( f: X \rightarrow Y \) of degree \( n \) and divisor \( D = \sum n_{i} P_{i} \) on \( X \), define a homomorphism \( f_{*} : \) Div \( X \rightarrow \) Div \( Y \) by \( f_{*}(\sum n_{i} P_{i}) = \sum n_{i} f(P_{i})\).

02

Address part (a)

We need to show \( \operatorname{det}(f_{*} \mathscr{L}(D)) \cong (\operatorname{det} f_{*} c_{X}) \otimes \mathscr{L}(f_{*} D) \). Consider an effective divisor \( D \) and apply \( f_{*} \) to the exact sequence \( 0 \rightarrow \mathscr{P}(-D) \rightarrow \varphi_{x} \rightarrow \mathscr{C}_{D} \rightarrow 0 \) . With respect to the properties of effective divisors and properties of determinant of a morphism, one can arrive at desired relationship.

03

Address part (b)

For this part, we need to show that \( f_{*} D \) only depends on the linear equivalence class of \( D \). Hence, a homomorphism \( f_{*} : \operatorname{Pic} X \rightarrow \) Pic \(Y\) is created . Also, since \( f_{*} \cdot f^{*} : \operatorname{Pic} Y \rightarrow \operatorname{Pic} Y \) , it follows that this is just multiplication by \( n \). This can be shown by applying properties of linear equivalence in algebraic geometry.

04

Address part (c)

This part deals with duality for a finite flat morphism and requires the demonstration of \( \operatorname{det} f_{*} \Omega_{X} \cong \text{ (det } f_{*}(x))^{-1} \otimes \Omega_{Y}^{ \otimes n} \) . This can be established using results from duality theory and properties of determinants of morphisms.

05

Address part (d)

With the assumption that \( f \) is separable, we have to demonstrate \( (\operatorname{det} f_{*} C_{x})^{2} \cong \mathscr{L}(-B) \). We first define the branch divisor \( B \) to be the divisor \( f_{*} R \) on \( Y \). Applying the morphism properties and ramification divisor properties, we can derive the required equation.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Finite morphisms

Finite morphisms are a class of morphisms in algebraic geometry. These can be thought of as maps between varieties that behave somewhat like finite graph connections. In the context of this problem, we consider a finite morphism, denoted as \( f: X \rightarrow Y \), mapping one algebraic curve \( X \) onto another \( Y \), and having a specific degree \( n \).

Here's what it means:

• "Finite" refers to the pre-images of points in \( Y \) being finite sets.
• The "degree", \( n \), signifies the number of pre-images each generic point in \( Y \) has under \( f \).

This property is crucial when working with divisors as it implies that the pullback or pushforward of divisors can be computed by accounting for this finite number of pre-images. Overall, these morphisms help in systematically transferring information between two varieties.

Divisors in algebraic geometry

Divisors are formal sums of points on an algebraic curve that capture important geometric information, such as the presence of zeroes and poles of functions. For an algebraic curve \( X \), a divisor \( D \) is a finite sum \( \sum n_i P_i \), where each \( P_i \) is a point on \( X \) and \( n_i \) is an integer.

In algebraic geometry, we often need to "pushforward" a divisor from one curve to another through a morphism. For our finite morphism \( f : X \rightarrow Y \), the pushforward of a divisor \( D \) is given by \( f_{*}(\sum n_i P_i) = \sum n_i f(P_i) \). This operation takes the effective part of the divisor and projects it onto the target variety according to the morphism \( f \).

By looking at divisors in this way, we get insights into how functions on one curve relate to those on another, particularly when considering how line bundles transform under morphisms.

Sheaves and determinants

Sheaves are fundamental tools in algebraic geometry, providing a way to systematically keep track of local mathematical data across different regions of a variety. When dealing with morphisms, understanding how sheaves, particularly vector bundles, behave is crucial.

For a locally free sheaf \( \delta \) on \( Y \) with rank \( r \), the determinant \( \mathrm{det} \mathscr{B}=\wedge^r \mathscr{E} \) comes into play. This is because the determinant helps us summarize complex transformation properties, especially when deriving global information from local data.

When we have a morphism like \( f \), and sheaves on both \( X \) and \( Y \), we compute determinants to relate the structures on one to another. For instance, in our problem, demonstrating that \( \mathrm{det}(f_{*} \mathscr{L}(D)) \cong (\mathrm{det} f_{*} c_{X}) \otimes \mathscr{L}(f_{*} D) \) illustrates how complexity can be parceled into manageable, analogous parts.
In fact, this equality helps establish correlations between different schemes of divisors, explaining why determinants are indispensable in algebraic geometry.

Picard groups

The Picard group, often denoted as \( \mathrm{Pic}(X) \), is a central concept in algebraic geometry. It's a way of classifying line bundles over a variety \( X \) up to an equivalence relation, called linear equivalence.

These groups are significant for several reasons:

• They capture how divisors behave under linear equivalence.
• They allow us to gauge the transformation of divisors under a morphism.
• They help us understand the geometry of varieties in the context of line bundles.

In our exercise, the morphism \( f: X \rightarrow Y \) allows the creation of a homomorphism \( f_{*}: \mathrm{Pic}(X) \rightarrow \mathrm{Pic}(Y) \). This homomorphism reveals how divisors on \( X \) that are linearly equivalent transform into divisors on \( Y \).

Furthermore, proving that \( f_{*} \cdot f^{*}: \mathrm{Pic}(Y) \rightarrow \mathrm{Pic}(Y) \) acts as multiplication by \( n \), the degree of \( f \), unwraps deeper insights into how the Picard group preserves algebraic structure through this morphism.