Prove the characterization of the exceptional set in a blow-up construction by showing the equality of two set descriptions | Step-by-Step Solution

Hi there! I can see you're working on a fascinating problem in algebraic geometry about blow-ups and exceptional sets. This is advanced material, so let's break it down together!

What We're Solving:

You need to prove that two different descriptions of the exceptional set in a blow-up construction are actually the same set. This involves understanding how the blow-up map behaves and how it relates to the original variety defined by polynomials.

The Approach:

The key insight here is that we're dealing with set equality, which means we need to prove two inclusions: that each set is contained in the other. This is a classic "double inclusion" proof strategy. We'll also need to understand what the exceptional set represents geometrically - it's the "new" part that gets added when we blow up.

Step-by-Step Solution:

Step 1: Set up your definitions clearly

• Write out precisely what the exceptional set E is in your blow-up construction
• Define the inverse image of your variety under the blow-up map π
• Make sure you understand what "defined by polynomials" means in this context

Step 2: Understand the geometric picture

• The blow-up replaces a subvariety (often a point or subspace) with its projectivized normal cone
• The exceptional set is exactly this "replacement" part
• Think about how points in the blow-up relate to points in the original variety

Step 3: Prove the first inclusion (E ⊆ inverse image)

• Take an arbitrary point in the exceptional set
• Show that when you apply the blow-up map π, this point maps to something in your original variety
• This uses the definition of how the blow-up map works

Step 4: Prove the second inclusion (inverse image ⊆ E)

• Take a point in the inverse image of your variety
• Show that if this point isn't in the exceptional set, you get a contradiction
• This often involves analyzing what happens to the defining polynomials under the blow-up

Step 5: Use the polynomial conditions

• The polynomials defining your variety give you concrete algebraic conditions
• Show how these conditions force points to lie in the exceptional set
• This is where the "defined by polynomials" part becomes crucial

The Framework:

Opening approach: Start by clearly stating what you're proving and set up your notation consistently.

Body structure:

• Paragraph 1: Define all objects (blow-up, exceptional set, the variety)
• Paragraph 2: Prove first inclusion with clear logical steps
• Paragraph 3: Prove second inclusion, handling the polynomial constraints
• Paragraph 4: Conclude by explaining the geometric significance

Key proof techniques to use:

• Double inclusion for set equality
• Contradiction arguments where helpful
• Direct verification using polynomial conditions

Memory Tip:

Think of the exceptional set as the "footprint" of what was replaced in the blow-up. Any point that maps back to your original variety and wasn't there originally must be part of this footprint - that's why the two descriptions are the same!

The beauty of this result is that it gives you two ways to think about the same geometric object: one more abstract (exceptional set) and one more computational (inverse image with polynomial conditions).

You've got this! The key is to be very careful with your definitions and take the inclusion proofs one step at a time.