How to Analyze Irreducibility and Smoothness in Projective Morphisms

Hello! This is a beautiful and deep problem in algebraic geometry. Let's break it down together and understand what we're really investigating here.

What We're Solving:

We want to understand what happens to the fibers of a surjective projective morphism f: X → Y between normal varieties when we look at "very general" points in Y. Specifically, we're investigating whether these fibers are irreducible and smooth.

The Approach:

This problem combines several fundamental concepts in algebraic geometry. We'll use the fact that "bad" behavior (like reducibility or singularities) typically occurs only on proper closed subsets, while "good" behavior holds on dense open sets. The key insight is that normality of our varieties and the projective nature of our morphism give us powerful tools to control the geometry of fibers.

Step-by-Step Solution:

Step 1: Understand "very general fiber"

• A "very general" fiber means we're looking at f⁻¹(y) for points y in a countable intersection of dense open subsets of Y
• This is stronger than just "general" - it means we avoid ALL the "bad" loci simultaneously

Step 2: Set up the irreducibility question

• Consider the locus Z ⊂ Y where fibers are reducible
• We need to show Z is a proper closed subset (or empty)
• Use the fact that if f⁻¹(y) = C₁ ∪ C₂ with C₁, C₂ proper closed subsets, then this condition is "upper semicontinuous"

Step 3: Apply semicontinuity of fiber dimension

• Since f is projective and surjective between normal varieties, the dimension of fibers is semicontinuous
• The locus where fibers have dimension larger than the generic dimension forms a proper closed subset
• This helps control where "extra components" can appear

Step 4: Use generic flatness

• By generic flatness (which applies since we're in the projective case), there exists a dense open U ⊂ Y such that f is flat over U
• Over this open set, all fibers have the same dimension and nice properties

Step 5: Address smoothness using Bertini-type results

• The singular locus of fibers forms a constructible set
• By a generalization of Bertini's theorem for morphisms, the locus of singular fibers has smaller dimension than Y (under our hypotheses)
• Therefore, very general fibers are smooth

Step 6: Combine normality with projectivity

• Normality of X and Y ensures we don't have pathological behavior
• Projectivity gives us compactness properties that make semicontinuity arguments work
• Together, these force "bad" fibers to lie in proper closed subsets

The Answer:

For a surjective projective morphism f: X → Y between normal varieties:

• Very general fibers are irreducible: The locus of reducible fibers is contained in a countable union of proper closed subsets
• Very general fibers are smooth: The locus of singular fibers has dimension less than dim(Y), so very general fibers avoid singularities

The key is that both reducibility and singularities are "special" conditions that can only occur on proper closed subsets when we have the nice geometric conditions (normal varieties, projective morphism).

Memory Tip:

Think of it this way: "Normal + Projective = Nice behavior generically." The normality prevents pathological singularities, while projectivity gives compactness that makes semicontinuity work. Bad things can happen, but only on "thin" sets, so if you pick your point generically enough, you avoid all the problems at once!

This is a wonderful example of how the right geometric hypotheses ensure that pathological behavior is rare, and generic behavior is what we'd hope for!