Relative Base Locus: Coincidence With Cokernel Support?

Hey guys! Ever find yourself diving deep into the fascinating world of algebraic geometry, particularly when dealing with birational geometry? Today, we're tackling a juicy question that might make your head spin – but in a good way! We're going to explore the connection between the relative base locus and the support of the cokernel of a certain morphism. Buckle up, because this is going to be a fun ride!

Setting the Stage: The Players in Our Geometric Drama

Before we jump into the heart of the matter, let's make sure we're all on the same page with the key players in our geometric drama. We're dealing with a proper morphism $f\colon X\rightarrow S$ between normal varieties. Now, what does that all mean?

• Proper Morphism: Think of a proper morphism as a map that preserves topological niceness. More precisely, it's a morphism where the inverse image of any compact set is compact. In the context of algebraic geometry, properness is a crucial property that ensures certain constructions behave well.
• Normal Varieties: A variety is essentially a geometric object defined by polynomial equations. A normal variety is one that satisfies a certain 'smoothness' condition – technically, its local rings are integrally closed domains. This normality ensures that we don't run into too many singularities, making our lives easier.
• Cartier Divisor: A Cartier divisor $D$ on $X$ is, roughly speaking, a formal sum of subvarieties of codimension one, where we allow coefficients. Think of it like a recipe for cutting out pieces of our variety $X$. Cartier divisors are incredibly important for studying line bundles and linear systems, which play a central role in algebraic geometry.

So, we have our stage set: a proper map between well-behaved geometric objects, and a recipe for carving out pieces of our space. Now, let's introduce the main question that's been bugging us.

The Big Question: Do They Coincide?

The central question we're tackling is: Does the relative base locus coincide with $\operatorname{Supp}\operatorname{coker} (f^*f_*\mathcal{O}_X(D)\rightarrow\mathcal{O}_X(D))$? Sounds intimidating, right? Let's break it down. We need to understand what the relative base locus is and what the heck $\operatorname{Supp}\operatorname{coker} (f^*f_*\mathcal{O}_X(D)\rightarrow\mathcal{O}_X(D))$ represents.

Diving into the Relative Base Locus

First up, the relative base locus. What is it? The concept of a base locus arises when we consider a linear system of divisors. A linear system, in essence, is a family of divisors that are 'linearly equivalent'. Now, what does 'linearly equivalent' mean in this context? Two Cartier divisors $D$ and $D'$ are linearly equivalent over $S$, denoted $D hicksim_S D'$, if their difference $D - D'$ is the divisor of a rational function on $X$ that is 'trivial' along the fibers of $f$. This 'triviality' condition ensures that the linear equivalence respects the morphism $f$. Intuitively, we're saying that $D$ and $D'$ differ by something that doesn't change as we move along the fibers of $f$.

The base locus of a linear system is the set of points where all the divisors in the system intersect. It's the 'fixed' part of the linear system that doesn't move around as we vary the divisors within the system. Now, the relative base locus is a generalization of this concept, taking into account the morphism $f$. It's the locus where the linear system has base points 'relative to $S$'. Think of it as the set of points in $X$ where the sections of the corresponding line bundle become simultaneously zero when we pull them back to $S$.

The relative base locus is a crucial concept in birational geometry. It tells us where our linear system is 'degenerate' in a certain sense. Understanding the relative base locus is essential for resolving singularities, studying the geometry of fibrations, and many other important problems.

Decoding $\operatorname{Supp}\operatorname{coker} (f^*f_*\mathcal{O}_X(D)\rightarrow\mathcal{O}_X(D))$

Okay, now for the seemingly monstrous expression: $\operatorname{Supp}\operatorname{coker} (f^*f_*\mathcal{O}_X(D)\rightarrow\mathcal{O}_X(D))$. Let's dissect this piece by piece. This expression involves some fundamental concepts from sheaf theory, which is a powerful language for studying geometric objects.

• $\mathcal{O}_X(D)$: This is the invertible sheaf associated with the Cartier divisor $D$. Sheaves are like 'functions' that vary from point to point on our variety. Invertible sheaves, also known as line bundles, are particularly nice sheaves that play a crucial role in algebraic geometry. $\mathcal{O}_X(D)$ essentially encodes the information about the divisor $D$ in a sheaf-theoretic way. Sections of this sheaf can be thought of as functions with poles along $D$.
• $f_*$: This is the pushforward functor. Given a sheaf on $X$, the pushforward $f_*$ gives us a sheaf on $S$. Think of it as 'averaging' the sheaf on $X$ along the fibers of $f$. More formally, for an open set $U$ in $S$, the sections of $f_*\mathcal{O}_X(D)$ over $U$ are the sections of $\mathcal{O}_X(D)$ over $f^{-1}(U)$.
• $f^*$: This is the pullback functor. Given a sheaf on $S$, the pullback $f^*$ gives us a sheaf on $X$. It's the 'opposite' of the pushforward. Think of it as 'lifting' the sheaf from $S$ to $X$. More formally, the pullback $f^* \mathcal{F}$ of a sheaf $\mathcal{F}$ on $S$ is characterized by the property that its sections over an open set $V$ in $X$ can be obtained from sections of $\mathcal{F}$ over open sets in $S$ that 'map to' $V$ under $f$.
• $f^*f_*\mathcal{O}_X(D)$: This is the composition of the pushforward and pullback functors. We start with $\mathcal{O}_X(D)$ on $X$, push it down to $S$ using $f_*$, and then pull it back up to $X$ using $f^*$. This resulting sheaf on $X$ carries information about how $\mathcal{O}_X(D)$ behaves along the fibers of $f$.
• $f^*f_*\mathcal{O}_X(D)\rightarrow\mathcal{O}_X(D)$: This is a morphism of sheaves, a map between these sheafy functions. It's a natural map that arises from the adjunction between the pushforward and pullback functors. This morphism tells us how well the 'push-pull' operation $f^*f_*$ approximates the original sheaf $\mathcal{O}_X(D)$.
• $\operatorname{coker}$: This stands for cokernel. The cokernel of a morphism is a measure of 'what's left over' after the morphism has done its thing. In the context of sheaves, the cokernel of $f^*f_*\mathcal{O}_X(D)\rightarrow\mathcal{O}_X(D)$ tells us how much $\mathcal{O}_X(D)$ fails to be 'reconstructed' by the push-pull operation $f^*f_*$.
• $\operatorname{Supp}$: This stands for support. The support of a sheaf is the set of points where the sheaf is 'non-zero'. In our case, $\operatorname{Supp}\operatorname{coker} (f^*f_*\mathcal{O}_X(D)\rightarrow\mathcal{O}_X(D))$ is the set of points in $X$ where the cokernel is non-zero, meaning the points where the morphism $f^*f_*\mathcal{O}_X(D)\rightarrow\mathcal{O}_X(D)$ fails to be surjective.

So, putting it all together, $\operatorname{Supp}\operatorname{coker} (f^*f_*\mathcal{O}_X(D)\rightarrow\mathcal{O}_X(D))$ is the set of points in $X$ where the push-pull operation $f^*f_*$ doesn't perfectly reconstruct the sheaf $\mathcal{O}_X(D)$. It's a measure of how much information is 'lost' when we push down to $S$ and then pull back up to $X$.

The Million-Dollar Question, Answered?

Now that we've deciphered the meaning of both the relative base locus and $\operatorname{Supp}\operatorname{coker} (f^*f_*\mathcal{O}_X(D)\rightarrow\mathcal{O}_X(D))$, we can finally address the central question: Do they coincide?

This is a deep question, and the answer, as is often the case in mathematics, is a bit nuanced. In general, the relative base locus and $\operatorname{Supp}\operatorname{coker} (f^*f_*\mathcal{O}_X(D)\rightarrow\mathcal{O}_X(D))$ do not always coincide. However, there are important cases where they do, and understanding when this happens is crucial for many applications.

There are theorems and results that explore this relationship under specific assumptions on $f$ and $D$. For instance, if $f$ is a projective morphism and $D$ is sufficiently ample, then we might expect a closer connection between the relative base locus and the support of the cokernel. The ampleness condition ensures that $D$ is 'positive' in a certain sense, which can help to control the behavior of the pushforward and pullback operations. Also, the geometry of the fibers of $f$ plays an important role. If the fibers are 'well-behaved', the relationship between the relative base locus and the cokernel becomes cleaner.

The intuition behind why they might coincide in some cases is that both the relative base locus and the cokernel measure a certain kind of 'degeneracy' related to the linear system associated with $D$. The relative base locus measures where the linear system has base points relative to $S$, while the cokernel measures where the push-pull operation fails to reconstruct the sheaf $\mathcal{O}_X(D)$. In situations where these two notions of degeneracy are closely related, we can expect the relative base locus and the support of the cokernel to coincide.

Why Does This Matter?

Okay, so we've explored this intricate relationship between the relative base locus and the support of a cokernel. But why should we care? What are the applications of this kind of result?

Well, understanding the connection between these two objects has significant implications in various areas of algebraic geometry, particularly in birational geometry. Birational geometry is concerned with studying varieties up to birational equivalence, meaning we consider varieties to be 'the same' if they have isomorphic open subsets. This allows us to focus on the 'essential' geometric properties of varieties, ignoring minor differences.

The relative base locus plays a crucial role in minimal model program (MMP), a powerful tool in birational geometry for classifying algebraic varieties. The MMP involves a series of birational transformations that aim to simplify a variety while preserving its essential geometric features. Understanding the relative base locus is essential for controlling these birational transformations and ensuring that they lead to a 'minimal model', a variety that is in some sense the 'simplest' representative of its birational equivalence class.

Furthermore, the cokernel $\operatorname{coker} (f^*f_*\mathcal{O}_X(D)\rightarrow\mathcal{O}_X(D))$ provides valuable information about the cohomology of the sheaf $\mathcal{O}_X(D)$. Cohomology is a powerful tool for studying the global properties of sheaves and varieties. By understanding the support of the cokernel, we can gain insights into the vanishing of certain cohomology groups, which has implications for the geometry of the variety and the linear system associated with $D$.

In essence, the question of whether the relative base locus coincides with $\operatorname{Supp}\operatorname{coker} (f^*f_*\mathcal{O}_X(D)\rightarrow\mathcal{O}_X(D))$ is not just an abstract curiosity. It's a question that lies at the heart of many fundamental problems in algebraic geometry and has deep connections to the classification of algebraic varieties.

Final Thoughts

So, guys, we've journeyed through a fascinating landscape of proper morphisms, normal varieties, Cartier divisors, relative base loci, and cokernels of sheaf morphisms. We've explored the intricate relationship between the relative base locus and $\operatorname{Supp}\operatorname{coker} (f^*f_*\mathcal{O}_X(D)\rightarrow\mathcal{O}_X(D))$, and we've seen why this question matters in the broader context of algebraic geometry.

While the answer to our central question isn't a simple 'yes' or 'no', the exploration has given us a deeper appreciation for the interplay between different concepts in algebraic geometry. The connection between the relative base locus and the support of the cokernel is a testament to the beautiful and intricate nature of this field, where seemingly disparate ideas often intertwine in surprising and profound ways. Keep exploring, keep questioning, and keep diving deeper into the amazing world of algebraic geometry!