Lecture I

1.1 Cardinality of finite sets vs Euler characteristic vs Chern-Schwartz-MacPherson classes

LetFindenote the category of finite sets I want to consider a functorCfromFin to abelian groups, defined as follows: forS a finite set,C(S) denotes the group of functionsS →Z.

Note: we could see C(S) as the group of linear combinations

V m V 1 V , whereV runs over the subsets of S,m V ∈Z, and1 V is the constant 1 onV and

0 in the complement ofV We could even select V to be the singletons{s}, with s∈S, if we wanted.

How do we make C into a functor? For f : S →T a map of finite sets, we have to decide whatC(f) does; and for this it is enough to decide what function

C(f)(1 V ) should be, for every subsetV ⊂S; and for this, we have to decide the value of

C(f)(1 V )(t) fort∈T Here is the definition:

The counting function C(f)(1 V)(t) is defined as the number of elements in the intersection of the preimage of t under f and the set V, represented mathematically as C(f)(1 V)(t) = #(f − 1(t) ∩ V) This simple observation leads to intriguing properties of the counting function, highlighting its role as a functor Notably, when applied to a singleton set {p}, the function yields C({p}) = Z, particularly in the context of the constant map κ: S → {p}.

So ifS 1, S 2 are two subsets ofS and S=S 1 ∪S 2, thinking about the covariance for

#(S 1 ∪S 2 ) = #S 1 + #S 2 −#(S 1 ∩S 2 ); and, more generally, the ‘inclusion-exclusion’ counting principle follows.

The topological Euler characteristic exhibits notable properties, particularly when applied to CW complexes For a space S structured as a CW complex, the Euler characteristic χ(S) is defined as the number of vertices minus the number of edges plus the number of faces This leads to an important relationship where, for any CW complexes S1, S2, and S, it holds that χ(S1 ∪ S2) = χ(S1) + χ(S2) - χ(S1 ∩ S2) This inclusion-exclusion principle for the Euler characteristic allows us to conceptualize it as a counting function.

The Chern-Schwartz-MacPherson (CSM) class of a variety V represents an advanced concept in this philosophical framework, functioning as a sophisticated form of 'counting' that adheres to the inclusion-exclusion principle Notably, the Euler characteristic, χ(V), of a compact complex algebraic variety V is intrinsically linked to the CSM class, as it corresponds to the degree of the zero-dimensional component of c SM (V) This CSM class is situated within a homology theory applicable to the variety V.

My emphasis will be: how do weconcretely compute such classes?

But maybe the first question should be: what does ‘computing’ mean?

The objection that we cannot compute SM (V) without a prior definition is valid; however, I have already defined the topological Euler characteristic Therefore, it is reasonable to inquire whether we can compute it now The definition itself, expressed as 'vertices minus edges plus faces,' serves as a form of computation.

That depends How would one use this in practice to compute the Euler characteristic of the subscheme ofP 3 given by the ideal

The concept of 'computation' is heavily influenced by the initial information available For instance, if I begin with a description of a variety \( V \) that allows for straightforward triangulation, calculating the Euler characteristic becomes simple However, as an algebraic geometer, I may need to work from more fundamental data, such as a defining homogeneous ideal in projective space This raises the question of how to derive a CW-complex realization of a scheme's support from its ideal In this context, the formula \( \chi(V) = \#\text{vertices} - \#\text{edges} + \ldots \) lacks true computational value; if I already possess sufficient information about \( V \) to count its vertices and edges, applying this formula does not provide significant new insights.

By contrast,here is what a computation is: themis{aluffi}1: Macaulay2

Singular-Factory 1.3b, copyright 1993-2001, G.-M Greuel, et al. Singular-Libfac 0.3.2, copyright 1996-2001, M Messollen i1 : load "CSM.m2"

loaded CSM.m2 i2 : QQ[x,y,z,w]; i3 : time CSM ideal(x^2+y^2+z^2,x*y-z*w)

used 49.73 seconds this tells me that the Chern-Schwartz-MacPherson class of that scheme is 4H 2 +

H 3 = 4[P 1 ] + [P 0 ] (once it is pushed forward into projective space); hence its Euler characteristic is 1.

In these lectures, my objective is to clarify the computation process by employing a 'divide and conquer' strategy This involves breaking down the information in the Chern-Schwartz-MacPherson class into two components: an 'easy' term known as the 'Chern-Fulton' class and an 'interesting' term referred to as the 'Milnor class,' which specifically addresses the singularities of the scheme.

‘Computing’ the Milnor class will be the most substantial part of the work.

To give an idea of how difficult it may be, here is a rather loose question:

Is there a natural scheme structure on the singularities of a given varietyV, which determines the Milnor class ofV?

To my knowledge, this is completely open! But it is understood rather well for hypersurfaces of nonsingular varieties, so that will be my focus in most of my lectures.

1.3 Chern-Schwartz-MacPherson classes: definition

In our analogy of counting, we define \( C(V) \) as the abelian group of finite linear combinations of closed subvarieties \( W \) of a variety \( V \), where the coefficients \( m \) are integers and the function \( 1_W \) equals 1 on \( W \) and 0 elsewhere The elements of \( C(V) \) are referred to as 'constructible functions' on \( V \) The question arises on how to structure \( C \) as a functor.

Forf :V 1 →V 2 a proper function (so that the image of a closed subvariety is a closed subvariety), it suffices to defineC(f)(1 W ) for a subvarietyW ofV 1 ; so we have to prescribeC(f)(1 W )(p) forp∈V 2 We set

The equation C(f)(1 W)(p) = χ(f − 1(p) ∩ W) establishes a functorial relationship in analogy with counting Additionally, there are other functors from the category of varieties and proper maps to abelian groups, notably the Chow group functor denoted by A A(V) is derived from C(V) by nullifying m W 1 W when a subvariety U of V exists with a rational function ϕ on U, where the divisor of ϕ equals m W [W] The subgroup generated by these constructible functions leads to the concept of ‘rational equivalence,’ defining A(V) as the abelian group of ‘cycles modulo rational equivalence.’ For proper maps f: V1 → V2, A(f)([1 W]) is defined as d[1 f(W)], where d represents the degree of f| W The Chow group A(V) can be viewed as a form of ‘homology,’ supported by a natural transformation.

A →H ∗ ; in fact, A(V) =H ∗ (V) in many interesting case, e.g.,V =P n

By construction there is a mapC(V)→ A(V); but thefunctors Cand Ado not have so much to do with each other – this is easily seennot to be a natural transformation.

This naive recipe defines a natural transformation on the associated graded functors \( G_C \) and \( G_A \), derived from the evident filtration by dimension However, the challenge lies in whether this transformation can be lifted to a natural transformation between \( C \) and \( A \) in a straightforward manner The question arises: if such a lift exists, does it manifest in a particularly interesting way? Assuming the existence of a lift, we consider a homomorphism \( c^* : C(V) \rightarrow A(V) \) for all \( V \) that satisfies covariance, prompting us to explore the preliminary implications of this relationship.

Just as we did when we were playing with finite sets, consider the constant mapκ:V → {p} The covariance diagram would then say that c ∗ (1 V ) =χ(V).

The degree of the class c ∗ (1 V) must correspond to the Euler characteristic of V This concept may evoke familiarity, particularly when considering nonsingular varieties In such cases, there exists a canonical class defined on V that possesses the property of having its degree equal to χ(V).

Answer: c(T V)∩[V] – in words, the ‘total homology Chern class of the tangent bundle ofV’ By one of the many descriptions of Chern classes, c(T V)∩

[V] measures the number of zeros of a tangent vector field on V, counted with multiplicities; this isχ(V), by the Poincar´e-Hopf theorem.

So we could make an educated guess: maybe a natural transformationc ∗ does exist, with the further amazing property thatc ∗ (1 V ) equalsc(T V)∩[V] whenever

This was conjectured by Pierre Deligne and Alexandre Grothendieck, and an explicit construction ofc ∗ was given by Robert MacPherson ([Mac74]).

In his note (871), p 361 and following, Grothendieck elaborates on this conjecture and its broader context, contributing to the rich mathematical commentary found in part II of [ReS].

Definition c SM(V) :=c ∗ (1 V ) is the ‘Chern-Schwartz-MacPherson class’ ofV.

Chern-Schwartz-MacPherson classes serve as characteristic classes specifically designed for singular varieties, as they are applicable to all varieties and align with traditional characteristic classes for nonsingular varieties.

The definition of Chern-Schwartz-MacPherson classes is only partially effective for computations and fails to capture the complexities of MacPherson's construction Additionally, it does not address the nuances of the alternative method proposed by Marie-Hélène Schwartz, which actually predates MacPherson's work, though it does not tackle the functorial framework; refer to [BS81] and [Bra00] for more information.

MacPherson’s construction is obtained by taking linear combinations of an- other notion of ‘characteristic class’ for singular varieties, also introduced by MacPherson, and usually named ‘Chern-Mather class’.

To define the concepts, consider a variety V embedded in a nonsingular variety M Each smooth point v in V can be mapped to its tangent space T_v V, which is viewed as a subspace of T_v M This process results in a rational map from V to Grass(dim V, T M), where the closure of this mapping is significant.

The Nash blow-up of the variety V is denoted as V, featuring a projection ν to V and a tautological bundle T derived from the Grassmann bundle Given that T aligns with the tangent bundle TV where applicable, it is logical to define a characteristic class using the formula c Ma (V) = ν ∗ (c(T) ∩ [V]).

Lecture II

2.1 (Chern-)Fulton classes vs (Chern-)Fulton-Johnson classes

I would like to look at a simple example to explore the difference between the Chern-Fulton and Chern-Fulton-Johnson business.

Consider the planar triple point S with ideal I = (x 2 , xy, y 2 ) in the affine planeA 2 For such an object we can be most explicit.

First of all,I is dominated byk[x, y] ⊕ 3 : k[x, y]s, t, u →I→0, sendings→x 2 , t→xy,u→y 2 Tensoring byk[x, y]/I:

The kernel of the map defined by (k[x, y]/I)s, t, u → I/I² → 0 is computed to be (xt - ys, xu - yt) Consequently, Proj(Sym I/I²) is characterized by the ideal (xt - ys, xu - yt) within the product S × P² According to Macaulay2, performing a primary decomposition reveals that this corresponds to the intersection (x, y) and an embedded component, indicating that Proj(Sym I/I²) is reduced.

P 2 inside the (nonreduced) S×P 2 The Fulton-Johnson class is then 1 times the class of the point supportingS.

What about Chern-Fulton? To obtain the Segre class ofSinA 2 , blow-upA 2 alongS We can fit the blow-up inA 2 ×P 2 , and a bit of patience gives its ideal:

(xt−ys, xu−yt, su−t 2 ).

The key distinction between the Rees algebra of I and the Symmetric algebra lies in the presence of an extra generator This difference is particularly notable when comparing the Rees algebra computation to the Fulton-Johnson computation, which yields the Symmetric algebra The exceptional divisor of the blow-up is formed by the intersection of the blow-up with S×P², resulting in S×a conic in P², as revealed by primary decomposition analysis.

It is completely different: it was aP 2 before, it is a curve now Its degree is

2×2 = 4, so the Chern-Fulton class is 4 times the class of the point.

Removing the extra generator from the ideal results in the multiple point (x², y²) becoming a complete intersection This alteration does not affect the Chern-Fulton class, as the ideal remains integral over the defined parameters Additionally, for complete intersections, the Chern-Fulton and Chern-Fulton-Johnson classes coincide, indicating that both will yield four times the class of the point in this scenario An exercise is provided to verify this conclusion explicitly.

One moral to be learned from such examples is that classes such as Chern- Fulton or Chern-Fulton-Johnson are extremely sensitive to thescheme structure.

I am focused on creating a tool capable of performing computations based on any arbitrary ideal However, the current discourse on Chern-Schwartz-MacPherson and Chern-Mather classes has not yet incorporated any scheme considerations.

The Chern-Schwartz-MacPherson class of a potentially non-reduced scheme is defined as the class of its support, which proves to be computationally advantageous, although it does not stem from any functoriality principles.

2.2 Segre classes again: inclusion-exclusion

Given the factors that have led us to this point, we should now consider whether classes like Chern-Fulton or Chern-Fulton-Johnson represent an advanced interpretation of

‘counting’ ? That is, do they satisfy ‘inclusion-exclusion’ ?

Since the difference between these classes and Segre classes is a common factor (c(T M)∩), the question for Fulton classes is equivalent to: do Segre classes satisfy inclusion-exclusion?

Example.TakeV = the union of two distinct linesL 1 ,L 2 in the projective plane. Thens(V,P 2 ) = (1+2H) 1 ∩[V] = [L 1 ]+[L 2 ]−4[pt], with hopefully evident notations.

On the other hand,s(L i ,P 2 ) = [L i ]−[pt] Thus s(L 1 ,P 2 ) +s(L 2 ,P 2 )−s(V,P 2 ) = 2[pt] =s(L 1 ∩L 2 ,P 2 ).

In other words, inclusion-exclusion fails miserably for Segre classes, on the very first example one may try.

There is indeed a solution that may seem trivial and was overlooked for many years It is noteworthy that variations of the definition of Segre classes generally adhere to the principle of inclusion-exclusion.

This follows immediately from 8th grade algebra The simplest case, which is also the only one I need in what follows, goes like this:

In the context of effective Cartier divisors \(X_1\) and \(X_2\) within an ambient scheme \(M\), the intersection \(Y = X_1 \cap X_2\) is defined scheme-theoretically By performing a blow-up of \(M\) along \(Y\) and designating \(E\) as the exceptional divisor, we find that the Segre classes satisfy the equation \(s(Y, M) = p^* 1 + E [E]\), where \(p\) is the blow-up map The pullback \(p^* X_i\) comprises \(E\) and a proper transform \(R_i\), with the property that \(R_1 R_2 = 0\) due to the non-intersecting nature of the proper transforms According to the projection formula, we can express the relationship as \(s(Y, M) = s(X_1, M) + s(X_2, M) - p^*[R_1] + [R_2] + [E]\), highlighting the algebraic connection between these Segre classes.

1 +R 1+R 2+E : something very close to inclusion-exclusion The funny term p ∗ [R 1] + [R 2] + [E]

1 +R 1 +R 2 +E is where one would expect ‘the class of the union’; this would instead equal p ∗ [R 1] + [R 2] + 2[E]

In my view, the inclusion-exclusion principle effectively applies to Segre classes when properly adjusted for the multiple contributions of subsets This concept can be generalized to accommodate an arbitrary number of subsets, enhancing its applicability and accuracy in mathematical contexts.

X i of any kind ) is one way to correct the classes We will run into another one later on.

Before massaging the formula we just obtained into something yielding any read- able information, I must introduce two simple notational devices.

These concern rational equivalence classes in an ambient scheme M, which

In this discussion, I will consider a pure-dimensional context, utilizing notations that index classes by codimension Specifically, I will denote a class A ∈ AN as A i ≥ 0 a i, where a i represents the portion of A with codimension i.

The term 'dual' of A is used to describe a concept where the notation may obscure the ambient space, potentially causing confusion This notation is straightforward: if E represents a vector bundle on M, or more broadly a class within the K-theory of vector bundles, it simplifies the representation of these mathematical structures.

The second piece of notation is similar, but a bit more interesting LetLbe a line bundle onM I will let

This also hides the ambientM; when necessary, I subscript the tensor:⊗ M But in these lectures all tensors will be in the ambient variety.

The rationale for the second notation is similar to the first: ifE is a class in

K-theoryand of rank0,then one can check that

Watch out for the ‘rank 0’ part!

Of course the notationA⊗ Lsuggests an ‘action’, and this is easy to verify: ifMis another line bundle, one checks that

All these observations are simple algebra of summations and binomial coeffi- cients; they are useful insofar as they compress complicated formulas into simpler ones by avoiding

Here is an example of such manipulations, which I will need in a moment.

Let’s go back to the ‘inclusion-exclusion’ formula we obtained a moment ago: s(Y, M) =s(X 1 , M) +s(X 2 , M)−p ∗ [R 1 +R 2 +E]

In this context, we define Y as the intersection of two hypersurfaces, X1 and X2, while E represents the exceptional divisor resulting from the blow-up along Y The proper transforms of X1 and X2 are denoted as R1 and R2, respectively Assuming that the ambient space M is nonsingular, we can express the Chern classes as cF(Y) = cF(X1) + cF(X2) - c(TM) ∩ p* [R1 + R2 + E].

1 +R 1 +R 2 +E, wherec Fdenotes Chern-Fulton class.

Now assume X 1 , X 2 , and Y are nonsingular Then c F = c SM, since both equal the classes of the tangent bundle That is: c SM(Y) =c SM(X 1) +c SM(X 2)−c(T M)∩p ∗ [R 1+R 2+E]

1 +R 1 +R 2 +E in this extremely special case On the other hand,c SMsatisfies inclusion-exclusion on the nose: c SM(Y) =c SM(X 1) +c SM(X 2)−c SM(X 1 ∪X 2).

The conclusion is that if X 1 , X 2 are transversal nonsingular hypersurfaces in a nonsingular ambient varietyM, and X=X 1 ∪X 2 , then c SM (X) =c(T M)∩p ∗ [R 1 +R 2 +E]

Let us work on the funny piece 1+R [R 1 +R 2 +E]

1 +R 2 +E in this formula First,R i +E stands for (the pull-back of)X i ; so we can rewrite this as 1+R [R 1 +R 2 +E]

1 +R 2 +E = 1+X [X − − E] E Next, another bit of 8th grade algebra:

Using the example from the previous section:

Put everything together and use the projection formula: c SM (X) =c(T M)∩p ∗

We have returned to the manifold M, where all aspects of the blow-up have been integrated into the original ambient space It is important to observe that the intersection of c(T M) with s(X, M) is contained within c F(X) Therefore, we can conclude that c SM(X) is equal to the sum of c F(X) and the intersection of c(T M).

We have proved that this holds ifX is the union of two nonsingular hypersurfaces in a nonsingular varietyM, meeting transversally alongY.

That is a reasonably pretty formula, but how do we interpret it in more ‘intrinsic’ terms? ‘What is’Y =X 1 ∩X 2, in terms ofX =X 1 ∪X 2, when X 1 and X 2 are nonsingular and transversal?

In a local context, let (F i) represent the ideal of X i, resulting in the ideal (F 1 F 2) for X The singularity subscheme of a hypersurface defined by the ideal (F) is characterized by the ideal (F, dF), where dF denotes the partial derivatives of F This scheme is inherently supported on the singular locus of X, and the specified ideal provides a coherent scheme structure that seamlessly integrates across affine overlaps.

For ourX, this ideal would be

The intersection of nonsingular varieties X₁ and X₂, denoted as Y = X₁ ∩ X₂, is supported by the independence of the differentials dF₁ and dF₂ at every point of Y, indicating that X₁ and X₂ are transversal Consequently, the ideal of the singularity subscheme is given by (F₁F₂, F₁, F₂) = (F₁, F₂), which corresponds exactly to the ideal of the intersection Y Thus, in this context, Y serves as the singularity subscheme of X.

Therefore, we can rephrase the formula we have obtained above: if X is a (very special) hypersurface in a nonsingular ambient variety M, and Y is the singularity subscheme ofX, then c SM (X) =c F (X) +c(T M)∩

Theorem 2.1 This formula holds foreveryhypersurface in a nonsingular variety.

This is the main result of the lectures: everything else I can say is simply a variation or restatement or application of this theorem.

Lecture III

I would like to review some of the approaches developed in order to prove the formula for the Milnor class of a hypersurfaceX in a nonsingular varietyM: c SM(X)−c F(X) =c(T M)∩ c(O(X)) − 1 ∩(s(Y, M) ∨ ⊗ O(X))

The initial study ([Alu99a]) is quite technical, yet it offers the benefit of being applicable to any algebraically closed field with a characteristic of 0, as well as to various hypersurfaces, including those with multiple components.

The objective is to demonstrate that the class defined by c ? (X) = c F (X) + c(T M) ∩ c(O(X)) - 1 ∩ (s(Y, M) ∨ ⊗ O(X)) exhibits covariance similar to that of the CSM class Achieving this would be a significant accomplishment, although I have yet to find a direct method to do so.

Using resolution of singularity, however, it is enough to prove a weak form of covariance, and this can be done Specifically:

• ifX is a divisor with normal crossings and (possibly multiple) nonsingular components, thenc ?(X) =c SM(X);

• if π : M→ M is a blow-up along a nonsingular subvariety of the singular locus ofX, thenπ ∗ c ?(π − 1 X) =c ?(X) +π ∗ c SM(π − 1 X)−c SM(X).

By resolving singularities, one can simplify the situation to the normal crossing case after performing several blow-ups Consequently, the properties of the canonical sheaf and the singularity map align at this stage, and as the resolution progresses, both the canonical sheaf and the singularity map exhibit consistent changes, ensuring their agreement for the variety X It is important to note that this process necessitates working with nonreduced objects The proofs of the two mentioned properties are quite technical, with the first case, concerning reduced divisors, ultimately demonstrating that if X is expressed as a union of reduced divisors with normal crossings and nonsingular components, the singularity subscheme Y leads to a specific relation involving s(Y, M).

⊗ L whereL=O(X) andL i =O(X i ) Proof: induction, and properties of⊗.

The other item is more interesting IfMis obtained by blowing up M along a nonsingularZ of codimensiond, one is reduced to showing that π ∗ c ?(π − 1 (X)) =c ?(X) + (d−1)c(T Z)∩[Z].

This should be much easier than it is! After a number of manipulations, one is led to transferring the question to within the bundleP(π ∗ P M 1 L ⊕ P M 1 L), where

The term "principal parts" (P 1) refers to two classes associated with the blow-up of manifold M along the singularity subscheme Y of variety X, and the blow-up of M along the singularity subscheme of π − 1(X) This distinction highlights the subtle differences between these two classes in the context of their respective blow-ups.

By framing the problem in this way, it becomes clear that the most effective approach is graph construction, a method highlighted in MacPherson's original paper This technique is specifically applied to the graph of the differential map π ∗ P M 1 L → P M 1 L, leading to the eventual proof of the necessary relation after addressing numerous technical details.

The proof I surveyed was developed between 1995 and 1996, but it wasn't published until 1999 due to unfavorable feedback from referees They suggested that an alternative perspective could lead to a more concise and insightful proof.

The accuracy of this approach has been confirmed, highlighting the importance of rephrasing the entire question using characteristic cycles This method translates the constructible function framework, which traces back to Claude Sabbah, and offers a robust solution A comprehensive discussion of this topic is presented in Lecture 3 of Jörg Schürmann’s lecture series.

Sabbah effectively encapsulates the situation by stating that the theory of Chern classes, as outlined by MacPherson, can be reduced to a Chow theory on the cotangent bundle T∗M, which solely involves fundamental classes Additionally, the functor of constructible functions is substituted with a functor of Lagrangian cycles.

The projectivization of the cotangent bundle T∗M enhances the geometric clarity of key operations on constructible functions, leading to a broader understanding of the theory For a comprehensive exploration of SM classes from this perspective, I recommend consulting [Ken90].

I will summarize the situation here LetMbe a nonsingular variety IfV ⊂M is nonsingular, then we have a sequence

The conormal bundle \( T V^* M \) is defined as a subvariety of the total space \( T^* M \), where \( V \) represents a singular variety To construct \( T V^* M \), one should first consider the nonsingular part of \( V \) and then close it in \( T^* M \) The linear combinations of cycles \( [T V^* M] \), referred to as the fundamental classes in Sabbah’s context, collectively form an abelian group denoted as \( L(M) \), with \( L \) signifying ‘Lagrangian’.

Now go back to the Nash blow-up ν : V → V, with tautological bundleT. For eachp∈V we can define a number as follows:

The 'local Euler obstruction' is a constructible function initially defined by MacPherson in a different context Notably, these functions form a basis for C(M), allowing us to establish a homomorphism, denoted as Ch, from C(V) to this space.

The characteristic cycle of a constructible function ϕ, denoted as Ch(ϕ), is defined as the cycle corresponding to ϕ, which satisfies the condition L(V): Ch(Eu V) = (-1)^dim V [T V ∗ M] Specifically, every subvariety V of M has a characteristic cycle, Ch(1 V), which is a combination of the conormal space to V and the tangent space T V ∗ M, taking into account the singularities of V.

The natural transformation \( c^* \), as originally defined by MacPherson, combines Chern-Mather classes with coefficients derived from local Euler obstructions Through a straightforward computation, it can be established that Chern-Mather classes are expressible in terms of conormal spaces The homomorphism \( C;L \) is designed to align with this framework Consequently, we obtain an explicit expression for \( c^* \): \[ c^*(\phi) = (-1)^{\text{dim } M - 1} c(TM) \cap \pi^* c(O(1)) - 1 \cap [PCh(\phi)] \]

, whereπis the projectionP(T ∗ M)→M ([PP01], p.67).

The operation described may initially seem unconventional, yet it represents the most straightforward method for handling classes within a projective bundle This process, referred to as casting the shadow of Ch(ϕ), reveals that the Chern-Schwartz-MacPherson class of V is essentially the shadow of its characteristic cycle.

In [PP01], Adam Parusiński and Piotr Pragacz present two proofs of a theorem that establishes the formula for the difference c SM(X) - c F(X), specifically for a hypersurface X within a nonsingular variety M, particularly in the reduced case.

Lecture IV

Once more, here is the main theorem of these lectures: ifX is a hypersurface in a nonsingular varietyM, then c SM (X) =c F (X) +c(T M)∩ c(O(X)) − 1 ∩(s(Y, M) ∨ ⊗ O(X))

The next natural question is: what about higher codimension? can one remove the hypothesis thatX be a hypersurface?

In a limited sense, the answer is affirmative, as indicated by the method involving differential forms with logarithmic poles, which applies to any arbitrary set X However, this approach offers little practical insight due to the lack of access to the required resolution for applying the formula Ideally, a statement should be formulated that remains confined within the context of M, similar to the primary formula discussed earlier.

The transition from hypersurfaces to complete intersections, or local complete intersections, is a crucial step in understanding Milnor classes, as highlighted by Jean-Paul Brasselet and further developed in Jörg Schürmann's comprehensive theory Schürmann's work builds upon the methods of Parusiński and Pragacz, focusing on 'interesting constructible functions' discussed in previous lectures This approach reinterprets the Milnor class as emerging from a constructible function linked to vanishing cycles, applicable to arbitrary complete intersections Our 'Main Theorem' indicates that for hypersurfaces, the information from vanishing cycles is primarily represented by the Segre class of the singularity subscheme, suggesting a need for further generalization.

Thus, a puzzle remains for higher codimensions, even in the cases covered by these approaches For this, I must go back to a question I posed in one of the first sections:

Is there a natural scheme structure on the singularities of a given varietyV, which determines the Milnor class ofV?

The Milnor class, differing only by sign, is defined as the difference between the stable manifold class SM(X) and the Fulton class c F(X) This definition aligns with the agreement of Fulton and Fulton-Johnson classes for local complete intersections, ensuring consistency with conventions used by other authors in this field.

The Milnor class of a hypersurface X is uniquely determined by its singularity subscheme, which is defined by the partial derivatives of a defining equation for X.

It is tempting to speculate that similar principles apply to broader varieties, with potential candidates for generalizing the 'singularity sub-scheme,' such as the base scheme of the rational map from V to its Nash blow-up However, there is currently no known formula that can initiate from this scheme, execute an intersection-theoretic operation akin to calculating the Segre class, and ultimately produce the Milnor class of V.

In this lecture, I will set aside the primary philosophical questions and instead employ a straightforward method to gather information on higher codi- mension without engaging in any additional work.

4.2 Segre classes: Inclusion-exclusion again

Brute force refers to revisiting the principles of inclusion-exclusion A modified definition of the Segre class introduces a concept that aligns with a specific form of inclusion-exclusion.

A variation of the Segre class that adheres to the principle of inclusion-exclusion can be established by redefining the primary formula for hypersurfaces within its definition This approach effectively integrates inclusion-exclusion into the framework of Segre classes.

Explicitly, define the SM-Segre class of a hypersurface X in a nonsingular varietyM, with singularity subschemeY, to be s ◦ (X, M) :=s(X, M) +c(O(X)) − 1 ∩(s(Y, M) ∨ ⊗ O(X)); and for a proper subschemeZ ofM, say

HereX i 1 ∪ ã ã ã ∪X i s is any hypersurfaces supported on the union.

This article highlights two key points regarding the formal exercise of verifying that s ◦ (Z, M) adheres to the inclusion-exclusion principle Specifically, it asserts that s ◦ (Z 1 ∩ Z 2, M) equals the sum of s ◦ (Z 1, M) and s ◦ (Z 2, M), minus s ◦ (Z 1 ∪ Z 2, M) for appropriate sets Z 1 and Z 2 The second point emphasizes the seemingly improbable nature of this definition, given the vast array of choices it encompasses It raises questions about the implications of altering the collection of hypersurfaces intersecting Z, such as incorporating additional inessential generators of the ideal of Z, and how these changes might affect the overall outcome.

I choose some other hypersurface supported onX i 1 ∪ ã ã ã ∪X i s ?

What magic property of Segre classes ensures that this definition does not depend on these choices?

I do not know! It must be something powerful indeed, but I have no clue as to its true nature.

So how do I know that the definition ofs ◦ (Z, M) really does not depend on the choices leading to it? Simply because c SM(Z) =c(T M)∩s ◦ (Z, M).

Exercise: realize that this is completely obvious, modulo the main result (The impatient reader may look up [Alu03b], Theorem 3.1.)

If I can compute ◦ (X, M) for hypersurfaces, I can extend this capability to all cases, effectively sidestepping the complexities associated with higher codimension This leads to a significant insight: the SM-Segre class ◦ (Z, M) is related to SM(Z) in a manner analogous to how the ordinary class s(Z, M) relates to the F(Z) class.

The qualifiers ‘almost ’ and ‘essentially ’ highlight a crucial distinction: the class ◦ (Z, M) is defined within the context of M, not Z Although there exists a class on Z that aligns with s ◦ (Z, M) after being pushed forward to M, I find it challenging to accurately map the class to its intended position This situation presents a further puzzle regarding the definition, which seems to be linked to a profound and enigmatic property of Segre classes.

Given these considerations, I must adjust my focus to calculating the push-forward of c SM (Z) to the ambient variety M, as this appears to be the most relevant question at this stage.

The algorithm underlying the Macaulay2 code demonstrated in the first lecture is now well understood If you can calculate Segre classes within a given ambient space M, you can utilize basic algebraic manipulations to compute the operation s ◦ (Z, M).

• obtain a set of hypersurfaces cutting out Z (for example, a homogeneous ideal forZ ifM =P n );

• for each subset S of this set, compute the singularity subschemeY S of the unionX S of the hypersurfaces inS;

• compute the ordinary Segre class ofY, and use it to obtains ◦ (X S , M);

• put all the information together and obtains ◦ (Z, M).

Obtainingc SM(Z) froms ◦ (Z, M) is trivial, as pointed out in the previous section.

In summary, the ability to compute ordinary Segre classes directly leads to the computation of Chern-Schwartz-MacPherson classes This realization brings us full circle to the initial challenge: determining how to compute Segre classes, a task that remains generally difficult, as highlighted in the first lecture.

The implementation of the algorithm operates within the framework of M = P^n, necessitating a reformulation of the primary hypersurface equation Let X represent a hypersurface in P^n; thus, O(X) is expressed as O(d + 1), where d equals deg X - 1, simplifying various related formulas If F is a homogeneous polynomial in k[z_0, , z_n] that defines X, the singularity subscheme Y of X is established by utilizing the ideal (∂z ∂F).

0 , , ∂z ∂F n ) (by Euler’s formula, this containsF) So we may seeY as a zero of a section

P n → O(d) n+1 : again our trick to realize a blow-up tells us what to do – the blow-up ofP n along

Y is the closure of the graph of the induced rational map

Lecture V

In this discussion, I will address several concrete questions that are loosely connected to the characteristic classes of singular varieties These inquiries aim to provide motivation and insight into the topic.

In the projective space \( P_N \) of plane curves of degree \( d \), where \( N = \frac{d(d+3)}{2} \), there exists a discriminant hypersurface \( D \) that encompasses all singular plane curves For any singular curve \( X \), it can be represented as a point on the hypersurface \( D \) The question arises regarding the multiplicity of the hypersurface \( D \) at the point \( X \).

If a space \( X \) lacks multiple components, its multiplicity equates to the sum of the Milnor numbers associated with \( X \) This relationship suggests a connection to Milnor classes and the characteristic classes of singular varieties.

It is in fact very easy to give a formula for this multiplicity The hypersurface

The locus D, with codimension 3, can be represented as the projection from P² × Pⁿ The equations defining this locus are derived from the three partial derivatives of a generic homogeneous polynomial of degree d in homogeneous coordinates.

P 2 Denoting by H, K respectively the hyperplane classes in P 2 and P N , each derivative is a polynomial of class (d−1)H+K, so the normal bundle ofD has class (1 + (d−1)H+K) 3

The multiplicity \( m_{X} D \) in the context of Segre classes is computed as \( s(X, D) = m_{X} D[X] \) Segre classes serve as birational invariants, expressed as \( s(X, D) = \pi^{*} s(\pi^{-1} X, D) \), where \( \pi: D \rightarrow D \) denotes the projection By capping with the classes of the ambient space, we obtain the intrinsic Fulton class, represented as \( c(T_{D}) \cap s(\pi^{-1} X, D) = c_{F}(\pi^{-1} X) \) This relationship can also be articulated as \( c(TP^{2} \times P^{N}) \cap s(\pi^{-1} X, P^{2} \times P^{N}) \), leading to the conclusion that \( s(\pi^{-1} X, D) = (1 + (d-1)H + K)^{3} \cap s(\pi^{-1} X, P^{2} \times P^{N}) \).

The singularity subscheme Y of X is represented by π − 1 X, which resides in a fiber P², ensuring that its Segre class remains independent of the Pⁿ factor This property is a result of the intrinsic nature of Fulton classes, and the operation K has no effect on it Consequently, the Segre class is expressed as s(π − 1 X, D) = (1 + (d−1)H)³ ∩ s(Y, P²).

The expression (1 + (d−1)H)³ represents the Chern class of the trivial extension of T∗P² tensored by L = O(d) This can be formulated as s(π−1X, D) = c(L)c(T∗M⊗L)∩s(Y, M), where M denotes P² The degree of this class corresponds to the multiplicity of the discriminant.

Claim This holds in general, for a hypersurface X in a nonsingular variety M, with singularity subschemeY.

The relationship between the computation of the general case and Milnor classes is evident, as detailed in [AC93] Specifically, the statement that \(s(\pi - 1 X, D) = s(Y, D)\) directly corresponds to the Milnor class, highlighting this significant connection.

⊗ L.Proof: Exercise Use the above formula for s(π − 1 X, D ) and the properties of ∨ and⊗to reduce to the main formula.

The relationship between the multiplicity of a discriminant and the parameters c SM and c F can be clearly articulated through a simple formula Discriminants inherently capture significant information regarding the characteristic classes of the entities they represent.

5.2 Constraints on singular loci of hypersurfaces

The class c(T ∗ M⊗ L)∩s(Y, M) contains significant information, as it represents the Milnor class through advanced algebraic concepts Initially, I did not recognize its connection to characteristic classes and independently explored it, referring to it as the ‘à-class of Y with respect to L’, denoted as à L (Y) While it is feasible to determine the precise dependence of this class on L, that aspect is not the focus of this discussion.

The Milnor class can be expressed in relation to the à-class, with the specific formula given by c SM(X)−c F(X) = c(L) dim X ∩ (à L (Y) ∨ ⊗ L) This indicates that much of the discussion can be directly translated into terms of Milnor classes.

The key feature of the à-class is its independence from the ambient variety M, meaning that the notation retains its significance regardless of the surrounding context Specifically, if you can represent Y as the singularity subscheme of a hypersurface with the bundle L (restricted to Y) in a different smooth ambient M, the computed à-class will remain consistent This consistency may not be immediately apparent, even after linking it to Milnor classes, raising questions about the transformations that X undergoes during this process.

In a unique scenario where the singularity subscheme Y of a variety X is nonsingular, it can be viewed as the singularity subscheme of the hypersurface defined by the equation 0 = 0 on Y This perspective allows us to utilize the main property of à-classes, which leads to the relationship c(T ∗ M ⊗ L) ∩ s(Y, M) = à L(Y) = c(T ∗ Y ⊗ L) ∩ [Y] This insight highlights the intriguing applications arising from the interplay between singularity subschemes and hypersurfaces in algebraic geometry.

One particularly pretty consequence of this formula is that it poses con- straints on ‘what’ nonsingular Y can be singularity subschemes of hypersurfaces in a givenM.

Example.The twisted cubic (with the reduced structure) cannot be the singularity subscheme of a hypersurface inM =P n

The twisted cubic is a curve of degree 3 embedded in projective space P^n, and it can be viewed as an abstract P^1 If it were the singularity subscheme of a hypersurface of degree d, the relationship would be expressed as c(T* P^n ⊗ O(X)) ∩ s(P^1, P^n) = O(3d)(P^1) = c(T* P^1 ⊗ O(3d)) ∩ [P^1] Here, h denotes the hyperplane in P^1.

1 + 3dh ∩[P 1 ] and finally (take degree 1 terms):

(3d−6)n−4 = 3d−2 which is nonsense (read modulo 3).

Numerous instances illustrate that a singularity subscheme is unlikely to be nonsingular A typical example, however defined, is expected to contain embedded components From a topological perspective, a general singular hypersurface is anticipated to exhibit a complex Whitney stratification, although explicit results in this area remain unclear.