Problem 6 - Algebraic Geometry

Question

(a) If \(C\) is a curve of genus \(g\), show that the diagonal \(\Delta \subseteq C \times C\) has self-intersection \(\Delta^{2}=2-2 g .\) (Use the definition of \(\Omega_{C / k}\) in (II, \(\S 8\) ).) (b) Let \(l=C \times\) pt and \(m=\mathrm{pt} \times C .\) If \(g \geqslant 1,\) show that \(l, m,\) and \(\Delta\) are linearly independent in \(\mathrm{Num}(C \times C) .\) Thus \(\mathrm{Num}(C \times C)\) has rank \(\geqslant 3,\) and in particular, \(\operatorname{Pic}(C \times C) \neq p_{1}^{*}\) Pic \(C \oplus p_{2}^{*}\) Pic \(C .\) Cf. (III, Ex. 12.6 ), (IV, Ex. 4.10 ).

Step-by-Step Solution

Answer

The self-intersection of the diagonal \(\Delta\) of \(C \times C\) is \(2 - 2g\). For\(g \geqslant 1\), the elements \(l\), \(m\), \(\Delta\) are linearly independent in \(\mathrm{Num}(C \times C)\), implying its rank is at least 3. Hence, \(\operatorname{Pic}(C \times C) \neq p_{1}^{*}\) Pic \(C \oplus p_{2}^{*}\) Pic \(C\).

Step 1: Determination of self-intersection

For the first part, we have to determine the self-intersection of the diagonal. Since the self-intersection of the diagonal \( \Delta \) of \( C \times C \) can be determined by \(\Delta . \Delta = \Delta^2\), we seek to show it equals \( 2 - 2g \). This claim can be proven using the arithmetic genus formula and the fact that \( \Delta \) is a copy of \( C \). Thus, \( \Delta^2 = \chi(\mathcal{O}_C) - 1 = 2 - 2g\) as per the definition given in (II, \(\S 8\)).

Step 2: Analyzing Linear Independence of \(l\), \(m\), \(\Delta\)

Moving to the second part, we analyze \(l\), \(m\), \(\Delta\). Here \(l = C \times \text{pt}\), \(m = \text{pt} \times C\) and \(\Delta\) is the diagonal. By showing that their intersection numbers are pairwise non-zero, we can ascertain their linear independence in \(\mathrm{Num}(C \times C)\), provided \(g \geqslant 1\). Note that \(l.\Delta = m.\Delta = g\) and \(l.m = 1\). Since \(g \geqslant 1\), neither of these products is zero, implying \(l\), \(m\), \(\Delta\) are linearly independent in \(\mathrm{Num}(C \times C)\). Thus, the rank of \(\mathrm{Num}(C \times C)\) is \( \geqslant 3\).

Step 3: Drawing Conclusion on \(\operatorname{Pic}(C \times C)\)

As a consequence of the linear independence of \(l\), \(m\), \(\Delta\), it can be established that \(\operatorname{Pic}(C \times C) \neq p_{1}^{*}\) Pic \(C \oplus p_{2}^{*}\) Pic \(C\). This follows from the calculation of the intersection products in Step 2 and by using the references given in the problem statement (III, Ex. 12.6), (IV, Ex. 4.10).

Key Concepts

Self-intersection

In algebraic geometry, self-intersection is an important concept that deals with the way a curve intersects with itself. For a curve \( C \) on a surface like \( C \times C \), the self-intersection of a diagonal \( \Delta \) refers to the way it intersects itself when considered on that surface. To find the self-intersection number \( \Delta^2 \), we use tools from the theory of divisors.

• The concept can be connected to the arithmetic genus formula, which provides a way to compute the self-intersection using properties like the genus \( g \) of the curve.
• For instance, using the arithmetic genus, \( \Delta^2 = 2 - 2g \). Here, \( \Delta \) is seen as a copy of the curve \( C \) within \( C \times C \).
• This calculation is supported by the mathematical structures that are part of algebraic geometry, revealing properties like the curve's genus \( g \) showing up in the formula.

Thus, understanding self-intersection helps in analyzing the deeper properties of algebraic surfaces and their constituent curves.

Linear Independence

Linear independence is a fundamental concept in both linear algebra and algebraic geometry, where it indicates a set of elements that do not overlap in a certain additive sense. In this exercise, we see linear independence come into play when dealing with elements \( l \), \( m \), and \( \Delta \) in the Num(C \times C) group.

• Elements \( l = C \times \text{pt} \) and \( m = \text{pt} \times C \) represent horizontal and vertical copies of the curve, while \( \Delta \) represents the diagonal.
• To determine their independence, one checks their intersection numbers: \( l.\Delta = g \), \( m.\Delta = g \), and \( l.m = 1 \).
• Here, \( g \ge 1 \) ensures these intersection numbers are nonzero, thereby establishing that \( l \), \( m \), \( \Delta \) are linearly independent.

This shows why the rank of \( \mathrm{Num}(C \times C) \) is at least 3, adding to our understanding of the interaction and independent nature of these geometric objects.

Picard group

The Picard group, denoted as \( \text{Pic}(X) \) for a variety or scheme \( X \), is the group of line bundles on \( X \), or equivalently, the group of divisors modulo linear equivalence. It plays a vital role in algebraic geometry as it helps classify vector bundles and sheaves over a given space.

• The exercise highlights that \( \text{Pic}(C \times C) \) is not merely the direct sum of the Picard groups \( p_{1}^* \text{Pic} C \) and \( p_{2}^* \text{Pic} C \).
• Given the rank condition \( \geqslant 3 \) from the linear independence of \( l, m, \Delta \), it showcases the complexity and richness of \( ext{Pic}(C \times C) \) beyond simple constructs.
• This discovery implies intertwining relations between algebraic curves that are beyond mere products.

Ultimately, exploring the Picard group in this context reveals deeper algebraic structures and relationships that highlight the intricacy and beauty of algebraic geometry.