Quadratic types and the dynamic Euler number of lines on a quintic threefold

We provide a geometric interpretation of the local contribution of a line to the count of lines on a quintic threefold over a field k of characteristic not equal to 2, that is, we define the type of a line on a quintic threefold and show that it coincides with the local index at the corresponding zero of the section of Sym^5 S^* ->Gr(2, 5) defined by the threefold. Furthermore, we define the dynamic Euler number which allows us to compute the A^1-Euler number as the sum of local contributions of zeros of a section with non-isolated zeros which deform with a general deformation. As an example we provide a quadratic count of 2875 distinguished lines on the Fermat quintic threefold which computes the dynamic Euler number of Sym^5 S^* ->Gr(2, 5). Combining those two results we get that the sum of the types of lines on a general quintic threefold is 1445<1>+ 1430<-1>in GW(k) when k is a field of characteristic not equal to 2 or 5.


Introduction
On a general quintic threefold there are finitely many lines.While the number of complex lines on a general quintic threefold is always 2875, the number of real lines depends on the choice of quintic threefold.However, we get an invariant count, namely 15, if we count each line with an assigned sign [OT14,FK15].So the real lines on a general quintic threefold can be divided into two 'types', one contributes a (+1), the other a (−1) to the invariant signed count.In [FK21] Finashin and Kharlamov provide a geometric interpretation of these assigned signs.They generalize the definition of the (Segre) type of a real line on a cubic surface defined by Segre [Seg42] to the type of a real line on a degree (2n − 1)-hypersurface in P n+1 and define it to be the product of degrees of certain Segre involutions.In particular, they define the type of a real line l on a quintic threefold X as follows: Any pair of points r, s on the line with the same tangent space T := T s X = T r X in X, defines an involution i : l ∼ = P 1 R → P 1 R ∼ = l which sends p ∈ l to the unique point q with T ∩T p X = T ∩T q X.
To each involution they assign a (+1) if the involution has fixed points defined over R and a (−1) if it does not.The type of a real line is the product of the assigned signs.Their definition naturally generalizes to lines defined over a field k of characteristic not equal to 2: Let l be a line defined over k on a general quintic threefold X.Then there are 3 pairs of points on l which have the same tangent space.Let r, s be one of those pairs, that is T := T s X = T r X, and assume that the closed subscheme of r and s is defined over a finite field extension L of k.Then we get an involution i : l L → l L of the base change of l to L that sends a point p ∈ l L to the point q ∈ l L with T ∩ T p X = T ∩ T q X.This involution has fixed points defined over L( √ α) for some α ∈ L × /(L × ) 2 .We say α is the degree of the involution i. Choosing suitable reprentatives of the degrees of the involutions, the product of the three degrees yields a well-defined element in GW(k).
Definition 1.1.The type of the line l is the product of the degrees of the 3 involutions viewed as an element of GW(k).
Here GW(k) denotes the Grothendieck-Witt group of finite rank non-degenerate symmetric bilinear forms (see for example [Lam05, Chapter II] for the definition of GW(k)) which is generated by α for α ∈ k × where α is the class of the form k × k → k, (x, y) → αxy in GW(k).
Let Y be a smooth and proper scheme over a field k and E → Y a relatively oriented vector bundle of rank equal to the dimension of Y .Then a general section σ : Y → E of the bundle has finitely many zeros.
Kass and Wickelgren define the A 1 -Euler number e A 1 (E) to be the sum of local indices ind x σ at the zeros of σ where the local index ind x σ is the local A 1 -degree at the zero x in local coordinates and a trivialization of E compatible with the relative orientation of E in the sense of [KW21,Definition 21].
Let S → Gr(2, 5) be the tautological bundle over the Grassmannian of lines in P 4 .We show in Proposition 2.5 that the vector bundle E := Sym 5 S * → Gr(2, 5) is relatively orientable.Furthermore, dim Gr(2, 4) = 6 = rank E. So the A 1 -Euler number of E is well-defined.
Let X = {f = 0} ⊂ P 4 be a general quintic threefold.Then f defines a general section σ f of E by restricting the defining polynomial f to the lines in P 4 .The lines on X are the lines with f | l = 0, i.e., the zeros of the section σ f .Whence, the A 1 -Euler number of E is by definition the sum of local indices at the lines on a general quintic threefold.Our main result in the first half of this paper is the following.
Theorem 1.2.Let X = {f = 0} ⊂ P 4 be a general quintic threefold and let l ⊂ X be a k-line on X.The type of l is equal to the local index at the corresponding zero of the section σ f : Gr(2, 5) → E.
It follows that the A 1 -Euler number of E is equal to the sum of types of lines on a general quintic threefold Here, Tr k(l)/k denotes the trace form, that is the composition of a finite rank non-degenerate symmetric bilinear form with the field trace.
Having defined the type, we want to compute the A 1 -Euler number e A 1 (E), that is, sum up all the types of lines on a general quintic threefold.However, many 'nice' quintic threefolds are not general and define sections with non-isolated zeros.Yet, a general deformation of such a threefold contains only finitely many lines.We define the dynamic Euler number of a relatively oriented vector bundle π : E → Y over a smooth and proper scheme Y over k with rank E = dim Y , to be the sum of the local indices at the zeros of a deformation σ t of a section σ valued in GW(k((t))), that is, the A 1 -Euler number of the base changed bundle (E → Y ) k((t)) expressed as the sum of local indices at the zeros of σ t .By Springer's Theorem (Theorem 5.2) the dynamic Euler number completely determines the A 1 -Euler number e A 1 (E) ∈ GW(k).
There are infinitely many lines on X and the section σ F defined by the Fermat does not have any isolated zeros.For a general deformation X t = {F t = F + tG + t 2 H + • • • = 0} of the X Albano and Katz find 2875 distinguished complex lines on X which are the limits of lines the deformation [AK91].Their computation still works over a field k of characteristic not equal to 2 or 5 and adding up the local indices at the deformed lines l t on the deformation X t , we get The A 1 -Euler number e A 1 (E) is the unique element in GW(k) that is mapped to e dynamic (E) by i : GW(k) → GW(k((t))) defined by a → a , that is, Combining this result with Theorem 1.2 we get that where the sum runs over the lines on a general quintic threefold and k is a field of characteristic not equal to 2 or 5.
Our computation reproves this result without using this theory and gives a new technique to compute 'dynamic' characteristic classes in the motivic setting.Additionally, we get a refinement of his result.The local index is only defined for isolated zeros.However, we can define the 'local index' ind l σ Ft at a line on the Fermat that deforms with a general deformation defined by F t to lines l t,1 , . . ., l t,m , to be the unique element in GW(k) that is mapped to ) where σ Ft is the section defined by restricting F t .
Theorem 1.4.Assume char k = 2, 5.There are well-defined local indices ind l σ Ft ∈ GW(k) of the lines on the Fermat quintic threefold that deform with a general deformation X t of the Fermat depending on the deformation and Note that i : GW(k) → GW(k((t))) is not an isomorphism, it is injective but not surjective.So it is not clear, that there exist such local indices in GW(k).This indicates that there should be a more general notion of the local index which can also be defined for non-isolated zeros of a section which deform with a general deformation.
Recall from [Ful98,Chapter 6] that classically the intersection product splits up as a sum of cycles supported on the distinguished varieties.We observe that this is true for the intersection product of σ F by the zero section in the enriched setting.
Theorem 1.5.The sum of local indices at the lines on a distinguished variety of the intersection product of σ F by s 0 that deform with a general deformation is independent of the chosen deformation.So for a distinguished variety Z of this intersection product, there is a well-defined local index ind Z σ F ∈ GW(k) and where the sum runs over the distinguished varieties.This is the first example of a quadratic dynamic intersection.
Kass and Wickelgren introduced the A 1 -Euler number in [KW21] to count lines on a smooth cubic surface.
Other related results are

The local index
Let X = {f = 0} ⊂ P 4 be a degree 5 hypersurface.Then f defines a section σ f of the bundle E := Sym 5 S * → Gr(2, 5) by restricting the polynomial f to the lines in P 4 .Here, S → Gr(2, 5) denotes the tautological bundle on the Grassmannian Gr(2, 5) of lines in P 4 .A line l lies on X if and only if f | l = 0 which occurs if and only if σ f (l) = 0.
Definition 2.1.Assume x is an isolated zero of σ f .The local index ind x σ f of σ f at x is the local A 1 -degree (see [KW19]) in coordinates and a trivialization of E compatible with a fixed relative orientation of E at x.
The A 1 -Euler number of E = Sym 5 S * → Gr(2, 5) is by definition [KW21] the sum of local indices at the zeros of a section σ f with only isolated zeros In other words, the A 1 -Euler number of E is a 'quadratic count' of the lines on quintic threefold X with finitely many lines.
Remark 2.2.The A 1 -Euler number is independent of the chosen section (given it has only isolated zeros) by [BW21, Theorem 1.1].
In this section we define a relative orientation of E and coordinates and local trivializations of E compatible with it.Then we give an algebraic description of the local index at a zero of a section defined by a general quintic threefold.

Relative orientability
We recall the definition of a relative orientation from [KW21, Definition 17].
Definition 2.3.A vector bundle π : E → X is relatively orientable, if there is a line bundle L → X and an isomorphism φ : Hom(det T X, det E) Here, T X → X denotes the tangent bundle on X.We call φ a relative orientation of E.
Remark 2.4.When T X and E are both orientable (that is, both are isomorphic to a square of a line bundle), then E is relatively orientable.However, the example of lines on a quintic threefold shows, that there are relatively orientable bundles which are not orientable.
Proposition 2.5.Denote by G the Grassmannian Gr(2, 5).The vector bundle E → G is relatively orientable.More precisely, there is a canonical isomorphism φ : Hom(det T G, det E) where Q denotes the quotient bundle on the Grassmannian G.
Proof.It follows from the natural isomorphism T G ∼ = S * ⊗Q that there is a canonical isomorphism det Thus,

Local coordinates
As in [KW21, Definition 4] we define the field of definition of a line l to be the residue field of the corresponding closed point in Gr(2, 5).We say that a line on a quintic threefold X = {f = 0} ⊂ P 4 is simple and isolated if the corresponding zero of the section σ f is simple and isolated.
Lemma 2.6.A line on a general quintic threefold X = {f = 0} ⊂ P 4 is simple and isolated and its field of definition is a separable field extension of k.
Proof.By [EH16, Theorem 6.34] the Fano scheme of lines on X is geometrically reduced and zero dimensional.
It follows that the zero locus {σ f = 0} is zero dimensional and geometrically reduced.It particular, it consists of finitely many, simple zeros.A line l on X with field of definition L defines a map Spec L → {σ f = 0} and Spec L is a connected component of {σ f = 0}.In particular, Spec L is geometrically reduced, which implies that l is simple and the field extension L/k is separable.
Let L be the field of definition of a line l on a general quintic threefold X = {f = 0} ⊂ P 4 and let (σ f ) L be the base changed section of the base changed bundle E L .By Lemma 2.6 L/k is separable.Recall that the trace form Tr L/k (β) of a finite rank non-degenerate symmetric bilinear form β over a finite separable field extension L over k is the finite rank non-degenerate symmetric bilinear form over k defined by the composition where Tr L/k denotes the field trace.Denote by (σ f ) L and Gr(2, 5) L be the base change to L and let Spec L Therefore, to compute the local index at l, we can assume that l ⊂ X is k-rational and take the trace form if necessary.
Remark 2.7.One can define traces in a much more general setting.Let A be a commutative ring with 1 and assume B is a finite projective A-algebra.After a coordinate change, assume that l = (0 : 0 : 0 : ).We define coordinates on the Grassmannian around l. Let e 1 , e 2 , e 3 , e 4 , e 5 be the standard basis of k5 .The line l is the 2-plane in k 5 spanned by e 4 and e 5 .Let U := A 6 = Spec k[x, x , y, y , z, z ] ⊂ Gr(2, 5) be the open affine subset of lines spanned by xe 1 + ye 2 + ze 3 + e 4 and x e 1 + y e 2 + z e 3 + e 5 .Note that the line l is the origin in U .
Remark 2.8.In general, a d-dimensional smooth scheme X does not have a covering by affine spaces and one has to use Nisnevich coordinates around a closed point x, that is an étale map ψ : U → A d for a Zariski neighborhood U of a closed point x such that ψ induces an isomorphism on the residue field k(x).Since ψ is étale, it defines a trivialization of T X| U where T X → X is the tangent bundle.Clearly, U = A 6 are Nisnevich coordinates around l.
Recall from [KW21, Definition 21] that a trivialization of E| U is compatible with the relative orientation φ : Hom(det T G, det E) Hom(det T G| U , det E| U ) sending the distinguished basis of det T X| U to the distinguished basis of det E| U , is sent to a square by φ.
Lemma 2.9.The restrictions T G| U and E| U have bases given by . The image of this distinguished basis element under φ defined in Proposition 2.5 is which is a square.In particular, the chosen trivialization of E| U is compatible with the relative orientation φ.
Proof.The canonical isomorphism det T G and the canonical isomorphism det Sym 5 It follows that φ sends the distinguished basis element to ( φ4 ∧ φ5

The local index at an isolated, simple zero of σ f
Let X = {f = 0} ⊂ P 4 be a quintic threefold and let l ⊂ X be an isolated, simple line defined over k.Let u, v, x 1 , x 2 , x 3 be the coordinates on P 4 .Again we assume that , that is, Q vanishes to degree at least two on l, and are homogeneous degree 4 polynomials in u and v.
where the entries of the matrix A are coefficients of P 1 , P 2 and P 3 .
Proof.Note that l is 0 in the coordinates from subsection 2.2.So the local index at l is the local A 1 -degree at 0 in the chosen coordinates and trivialization of E. Since l is simple, that is, 0 is a simple zero in the chosen coordinates and trivialization, it follows from [KW19] that the local A 1 -degree at 0 is equal to the determinant of the jacobian matrix of σ f in the local coordinates and trivialization defined in Lemma 2.9 evaluated at 0.
The section σ f in those local coordinates and trivialization is the morphism (f 1 , . . ., f 6 ) : A 6 → A 6 defined by the 6 polynomials which are the coefficients of u 5 , u 4 v, . . ., v 5 in Since Q vanishes to at least order two on the line, the partial derivative of Q in any of the six directions x, x , y, y , z, z evaluated at 0 vanishes.So the partial derivative that is, the first column of (4).Similarly, one computes the remaining columns of (4).
Remark 2.11.We have seen in Lemma 2.6 that on a general quintic threefold, all lines are isolated and simple.An isolated line l is not simple if and only if the derivative of σ f at l vanishes if and only if det A P1,P2,P3 = 0.In the case that l is isolated but not simple, one can use the EKL-form [KW19] to compute the corresponding local index.

Definition of the type
In this section we provide the definition of the type of an isolated, simple line.For the definition of the type of a line, we need to work over a field of characteristic not equal to 2 because there are involutions involved.Again we assume by base change that k(l) = k and that l = Recall from section 2.3 that under these assumptions the polynomial f Let C : l → P 2 be the degree 4 rational plane curve (u : v) → (P 1 (u, v) : P 2 (u, v) : P 3 (u, v)).Then C has the following geometric description: The 3-planes in P 4 containing l can be parametrized by a P 2 .Identify the tangent space T p X at a point p ∈ l in X with a 3-plane in P 4 .Then the corresponding 4-plane in k 5 has normal vector (P 1 (p), P 2 (p), P 3 (p), 0, 0).Therefore, C maps a point p ∈ l to its tangent space T p X in P 4 , i.e., C is the Gauß map.By the Castelnuovo count (see e.g.[ACGH85]), a general degree 4 rational plane curve has 3 double points.Furthermore, Finashin and Kharlamov show that the number of double points on C, given it is finite, is always 3 (possibly counted with multiplicities) (see [FK21, Proposition 4.3.3]).However, there could also be infinitely many double points on C. We will deal with this case in 4.2.3.For now we assume that C has 3 double points.That means that there are 3 pairs of points (r i , s i ) on the line l which have the same tangent space in X, i.e., T ri X = T si X for i = 1, 2, 3.
Let M be one double point of the curve C with field of definition (= the residue field of M in P 2 ) L M and let D be the corresponding degree 2 divisor on l.Let H t be the pencil of lines in P 2 through M .Then t defines a pencil of degree 4 divisors on l.
Lemma 3.1.When l is a simple line, the residual pencil D r t of degree 2 divisors on l is base point free, that means, there is no point on l where every element of the pencil D r t vanishes.
Proof.This follows from [FK21, Lemma 4.1.1 and §5.1].However, we reprove the result in our notation and setting.Without loss of generality, we can assume that M = (1 : 0 : 0).That means, there are two points on the line l that are sent to (1 : 0 : 0) by C. Let Q be the homogeneous degree 2 polynomial in u and v that vanishes on those two points.It follows that Q divides both P 2 and P 3 .Let t had a basepoint, then Q M 1 and Q M 2 would have a common factor and thus P 2 and P 3 had a common degree 3 factor and there would be nonzero degree 1 homogeneous polynomials r 2 and r 3 in u and v such that r 2 P 2 + r 3 P 3 = 0 • P 1 + r 2 P 2 + r 3 P 3 = 0.But then det A P1,P2,P3 = 0 by Lemma 2.12 where A P1,P2,P3 is the matrix from Proposition 2.10, that is, l is not a simple line.
Definition 3.2.We call the nontrivial element of the Galois group of the double covering Segre involution and denote it by Its fixed points are called Segre fixed points.
Figure 1 illustrates what the Segre involution i M does.Each element H t0 of the pencil H t of lines through M intersects the curve C in two additional points C(p) and C(q).The involution i M swaps the preimages p and q of those two points on l.
Geometrically, the Segre involution can be described as follows.
where r, s ∈ P 1 with C(s) = C(r) = M .Then to a point p ∈ l ⊗ L M there exists exactly one The involution i M swaps p and q.
The Segre fixed points of i M are defined over We say α M is the degree of the Segre involution i M .
Remark 3.3.The involution i M is a self map of the motivic sphere P 1 L M and thus, we can assign an A 1 -degree Lemma 3.4.The degree of the Segre involution i M : Proof.This follows from the proof of [KW21, Proposition 14].
We want to define the type of l to be the product of the three Segre involutions.We need a well-defined element of k × /(k × ) 2 in order to get a well-defined element of GW(k).If all three double points are defined over k, the degrees of the Segre involutions are elements of k × /(k × ) 2 , so the product is a well-defined element of k × /(k × ) 2 .If a double point is defined over a proper field extension L over k, its degree is an element of L × /(L × ) 2 and its contribution to the product of the degrees of the three involutions is only well-defined up to a square in L × which might not be a square in k × .However, we will see that the double points and their degrees come in Galois orbits.The product of Galois conjugate degrees is equal to the norm and lies in k × .
In particular, this is well-defined up to a square in k × .
Definition 3.5.Let M be the locus of double points of C and assume char k = 2.The type of the (isolated and simple) k-line l on X is where the product runs over the Galois orbits of M.
Remark 3.6.Recall that a line on a smooth cubic surface gives rise to a (single) Segre involution in a similar way: To each point on the line there is exactly one other point with the same tangent space and the Segre involution swaps those two points.In [KW21] Kass and Wickelgren define the type of a line on a smooth cubic surface to be the degree of this Segre involution.
Remark 3.7.If all three double points are pairwise different, the fixed points of the involutions i M form a degree 6 divisor on the line l which corresponds to an étale k-algebra A. Then the type of l is equal to the discriminant of A over k Recall that the type of a line on a smooth cubic surface is also the discriminant of the fixed point scheme of a Segre involution [KW21, Corollary 13].
4 The type is equal to the local index Theorem 4.1.Let X = {f = 0} ⊂ P 4 be a quintic threefold and l ⊂ X be a simple and isolated line with field of definition a field k of characteristic not equal to 2. The type of l is equal to the local index at l of σ f in GW(k).
Corollary 4.2.For X a general quintic threefold we have when k is a field with char k = 2.
This means that when we count lines on a general quintic threefold weighted by the product of the degrees of the fixed points of the Segre involutions corresponding to the three double points of the Gauss map of the line, we get an invariant element of GW(k).

An algebraic interpretation of the product of resultants
We have seen that the type of a line is equal to the product of resultants of quadratic polynomials in Lemma 3.4.We give an algebraic interpretation of the product of resultants for a special choice of the quadratic polynomials and will prove Theorem 4.1 by reducing to this case.
Proposition 4.3.Let Q 1 , Q 2 and Q 3 be homogeneous degree 2 polynomials in u and v and let where A P1,P2,P3 is the matrix defined in Proposition 2.10.
Proof.The Proposition can be shown by computing both sides of (7).However, we want to give another proof that illustrates what is going on.
We first show that det A P1,P2,P3 = 0 if and only if As both sides in (7) are homogeneous polynomials in the coefficients of the Q i , it follows that one is a scalar multiple of the other.We show equality for nice choice of Q 1 , Q 2 and Q 3 which implies equality in (7).
Assume that Res(Q 1 , Q 2 ) = 0. Then Q 1 and Q 2 have a common degree 1 factor and there are by Lemma 2.12.
Conversely, assume that det A P1,P2,P3 = 0 and there are r 1 , r 2 , r 3 ∈ k[u, v] 1 not all zero, such that Since deg r 1 = 1 and deg Q 1 = 2, either Q 2 and Q 1 share a degree 1 factor or Q 3 and Q 1 do (or both) and To show equality of (7) it remains to show equality for one choice of Q 1 , Q 2 and Q 3 such that det A P1,P2,P3 and Res(Q

Proof of Theorem 4.1
Let l be a simple and isolated line with field of definition k on a quintic threefold and let C = (P 1 : P 2 : P 3 ) : l → P 2 be the associated degree 4 curve defined in section 3.There are the following possibilities for how C : l → P 2 looks like.
1. C is birational onto its image.Then there are two possibilities (a) the generic case: the image has three distinct double points in general position.
(b) the curve has a tacnode, that is a double point with multiplicity two and another double point.
2. C is not birational onto its image.
A degree 4 map P 1 → P 4 which is not birational onto its image, is either a degree 4 cover of a line or a degree 2 cover of a conic.It was pointed out to me by the anonymous referee that if C were a degree 4 cover of a line, there would be linear r 1 , r 2 , r 3 ∈ k[u, v] such that r 1 P 1 + r 2 P 2 + r 3 P 3 = 0.In this case det A P1,P2,P3 = 0 as shown in Proposition 4.3, and the line line would not be simple and isolated.So in case C is not birational onto its images, it is a degree 2 cover of a conic.
We prove Theorem 4.1 case by case.

The generic case
In case C has three double points in general position, we perform a coordinate change of P 4 with the aim to being able to apply Proposition 4.3.We do this for all possible field extensions over which the three double points in M can be defined.There are three possibilities.
1.All three points in M are k-rational.
2. One point in M is rational and the other two are defined over a quadratic field extension L = k( √ β).
Since char k = 2, the field extension L/k is Galois.Let σ be the nontrivial element of Gal(L/k).
3. Let L = k(β) be a degree 3 field extension of k and assume that L the is residue field of one of the double points M .Because we assumed that the three double points are in general position and thus pairwise different, we know that L/k is separable.Let E/L be the smallest field extension such that E/k is Galois.Let G be the Galois group of E over k and let σ, τ ∈ G such that the other two double points of C are σ(M ) and τ (M ).
We choose three special points M 1 , M 2 and M 3 in P 2 for each of the three cases as in the following table. 1.
Claim 4.4.For each of the three casese, there is φ ∈ Gl 3 (k) that maps the three double points to M 1 , M 2 and M 3 . Proof.
1. Let (m 1 : m 2 : m 3 ), (n 1 : n 2 : n 3 ) and (q 1 : q 2 : q 3 ) be the three double points.In the first case, they are all k-rational and in general position.So the matrix is invertible.Its inverse is the map we are looking for.

Let (m
be the three double points with m i , n i , q i ∈ k for i = 1, 2, 3. Again the matrix invertible and its inverse sends the three double points to M 1 , M 2 and M 3 .

Let (m
m 3 + τ (β)n 3 + τ (β) 2 q 3 ) be the three double points.Again the map we are looking for is the inverse of     be the endomorphism of P 2 that maps the three double points to M 1 , M 2 and M 3 .
Claim 4.5.After a coordinate change we can assume that P 1 , P 2 and P 3 equal 1.
for some homogeneous degree Proof.Let s i , r i ∈ l ∼ = P 1 be the two points that are sent to M i by C and let ] 2 be a homogeneous degree 2 polynomial with zeros s i and r i for i = 1, 2, 3.
1.In case all double points are k-rational, we have 2. In case two of the double points are defined over the quadratic field extension 3. If there is no k-rational double point.For the degree 3 field extension L of k over which the first double point is defined, we have and

C has a tacnode
Assume that the double points of C are not in general position, that is they lie on a line.Because of Bézout's theorem, the three double points cannot be distinct in this case.Hence, there is a tacnode, that is a double point of multiplitcity 2. Note that, again by Bézout, there cannot be a double point of multiplicity 3. Since one of the double points has multiplicity 2, its field of definition could be non-separable of degree 2. However, we assumed that char k = 2, so this cannot be the case.So both double points are defined over k and we can assume that after a k-linear coordinate change the double point of multiplicity 1 is M 1 = (1 : 0 : 0) and the double point of multiplicity 2 is M 2 = (0 : 1 : 0).Let r 1 , s 1 ∈ P 1 be the two points that are sent to M 1 by C and Q 1 ∈ k[u, v] 2 be a homogeneous degree 2 polynomial that vanishes r 1 and s 1 .Further, let r 2 , s 2 ∈ P 1 be the two points that are sent to M 2 by C and Q 2 ∈ k[u, v] 2 be a homogeneous degree 2 polynomial that vanishes r 2 and s 2 .Then Q 1 divides P 2 and P 3 and Q 2 divides P 1 and P 3 .Since M 2 is a double point of multiplicity 2, we get that up to scalars in k The degree of i M1 is equal to Res(S, Q 2 ) and the degree of i M2 is equal to Res(Q 1 , Q 2 ).Computing both sides shows that det A P1,P2,P3 = Res(Q 1 , Q 2 ) 2 Res(S, Q 2 ) and thus Theorem 4.1 holds.

C is not birational onto its image
Finally, we deal with the case that C is not birational onto its image and C is a degree 2 cover of a conic, that is, C : l ∼ = P 1 → P 2 factors through a degree 2 map (Q 1 : Q 2 ) : where So, det A P1,P2,P3 equals the degree of the Segre involution corresponding to the degree 2 map (Q 1 : Remark 4.7.In this case, the type of l reminds of Kass and Wickelgren's definition of the type of a line on a cubic surface.For lines on cubic surfaces the Gauss map (P 1 : P 2 ) : l → P 1 has degree 2 and Example 4.8.Let P 1 = u 2 (u 2 + v 2 ), P 2 = u 2 v 2 and P 3 = v 2 (u 2 + v 2 ).Then C(t) = C(−t) for each t ∈ l ∼ = P 1 .Hence, C factors through (u 2 : v 2 ) : P 1 → P 1 and the corresponding Segre involution P 1 → P 1 is given by t → −t which has degree 1 ∈ k × /(k × ) 2 and one computes that det A P1,P2,P3 = 1 ∈ GW(k).
Remark 4.9.Let p be a singular point on a hypersurface Y = {h = 0}, such that the gradient grad h has an isolated zero at p.In [PW21, §6.3] Wickelgren and the author show that for a general deformation of Y the A 1 -Milnor number at p equals the sum of A 1 -Milnor numbers at the singularities which p bifurcates into.We can apply the same argument to our situation.Let l be an isolated not necessarily simple line on a quintic threefold X.If we deform the threefold ] for g a general homogeneous degree 5 polynomial, then the local index ind l σ f equals the sum of local indices at the lines the line l deforms to by [PW21, Theorem 5].We expect these deformations of l to be simple with a Gauss map with three double points (in general position) when the deformation is general, in which case ind l σ f would equal the sum of types of lines l deforms to.
local indices at the finitely many isolated zeros of σ.However, many 'nice' sections σ have non-isolated zeros and we have an excess intersection.In this section, we will use dynamic intersection to express e A 1 (E) as the sum of local contributions of finitely many closed points in σ −1 (0) that deform with a general deformation of σ.
Excess intersection of Grothendieck-Witt groups has already been defined and studied by Fasel in [Fas09] and Euler classes with support were defined in [Lev20, Definition 5.1] and further studied in [DJK21].Remark 5.20 in [BW21] shows that the contribution from a non-isolated zero which is regularly embedded, is the Euler number of a certain excess bundle.

Fulton's intersection product
Classically, we can define the Euler class e(E, σ) of a rank r vector bundle π : E → Y over a d-dimensional scheme Y over C with respect to a section σ as the intersection product of σ by the zero section s 0 [Ful98, Chapter 6].
Let C = C σ −1 (0) Y be the normal cone to the embedding s 0 : ).The intersection product of σ by the zero

and we define the
Euler class e(E, σ) with respect to σ to be this intersection product Let C 1 , . . ., C s be the irreducible subvarieties of C.
where x ) −1 ) and for x an isolated zero of a section σ of the relatively oriented bundle π : E → Y , the local index ind x σ computes e x (E, σ) as follows.Let ψ : U → Spec(k(x)[x 1 , . . ., x n ]) be Nisnevich coordinates around x and E| U ∼ = U × Spec(k(x)[y 1 , . . ., y d ]) a trivialization compatible with ψ and the relative orientation of E. Then by [Lev20,§5] where the xi are the images of the Nisnevich coordinates in m x /m 2 x .In other word the local index of an isolated zero agrees with this contribution described by Levine.
We conjecture that for any section σ of π : E → Y , not only sections with only isolated zeros, the A 1 -Euler number is the sum of 'local indices' at the distinguished varieties Z i as in the classical case (9).We will see that this is true in the case of the section σ F of E = Sym 5 S * → Gr(2, 5) defined by the Fermat quintic threefold {F = X 5 0 + X 5 1 + X 5 2 + X 5 3 + X 5 4 = 0} ⊂ P 4 , that is, we will show that there are well-defined 'local indices' at the distinguished varieties of the intersection product of σ F by the zero section s 0 .We then verify that the sum of these local indices is equal to e A 1 (E) in GW(k).The local index at a distinguished variety Z can be computed as follows.For each deformation of the Fermat we can assign local indices to the points in σ −1 F (0) ⊂ Gr(2, 5) that deform and the local index at Z is the sum of local indices at points in Z that deform with a general deformation.It turns out that the local index at Z is well-defined, that is, it does not depend on the chosen general deformation of the Fermat.

The dynamic Euler number
One way to find the well-defined zero cycle supported on the distinguished varieties classically is to use dynamic intersection [Ful98,Chapter 11].We deform a section σ of a rank E = dim Y bundle π : E → Y to σ t := σ + tσ 1 + t 2 σ 2 + . . .where the σ i are general sections of E. The deformation has finitely many isolated zeros and the zero cycle we are looking for is the 'limit' t → 0 of σ −1 t (0) which is supported on σ −1 (0) [Ful98, Theorem 11.2].Moreover, for a general deformation the zero cycle It follows that over the complex numbers the Euler number can be can be computed as the count of zeros of a section (with non-isolated zeros) that deform with a general deformation.For example, Segre finds 27 distinguished lines on the union of three hyperplanes in P 3 which deform with a general deformation [Seg42] and 27 is the classical count of lines on a general cubic surface [Cay09].Albano and Katz find the limits of 2875 complex lines on a general deformation of the Fermat quintic threefold in [AK91].In section 6 we will use those 2875 limiting lines to compute the 'dynamic Euler number' of Sym 5 S * → Gr(2, 5) valued in GW(k((t))).

GW(k((t)))
In order to understand the computations in section 6, we recall some properties of GW(k((t))).Any uni in k((t)) is of the form u = ∞ i=m a i t i with a m = 0.One can factor u as u = a m t m (1 Note that H := 1 + −1 = t + −t .Claim 5.1 illustrates the content of the following theorem. Theorem 5.2 (Springer's Theorem [Lam05]).

Definition of the dynamic Euler number
Let π : E → Y be a relatively orientable vector bundle with rank E = dim Y , Y smooth and proper over k, and let σ : Y → E be a section.We deform the section σ to σ t = σ + tσ 1 + t 2 σ 2 + . . .for σ i general sections of E. Then σ t is a general section of the base change E k((t)) to the field k((t)).In particular, σ t has only isolated zeros and the A 1 -Euler number e A 1 (E k((t)) ) ∈ GW(k((t))) is equal to the sum of local indices at those isolated zeros.
Definition 5.3.We call the dynamic Euler number of E.
By functoriality of the Euler class this sum (10) is in the image of the injective map i : GW(k) → GW(k((t))) from Springer's theorem 5.2.In other words, the A 1 -Euler number e A 1 (E) in is the unique element of GW(k) that is mapped to e dynamic (E) by i.We will see moreover how the local indices at the lines limiting to a distinguished variety of the intersection product of the Fermat section σ F by the zero section sum up to an element of GW(k) independent of the deformation, even though the local indices at these lines themselves depend on the deformation.
6 The lines on the Fermat quintic threefold Let X = {F = X 5 0 + X 5 1 + X 5 2 + X 5 3 + X 5 4 = 0} ⊂ P 4 be the Fermat quintic threefold.It is well known that there are infinitely many lines on X.In [AK91, §1] Albano and Katz show that the complex lines on X are precisely the lines that lie in one of the 50 irreducible components X ∩ V (X i + ζX j ) ⊂ P 4 where i, j ∈ {0, 1, 2, 3, 4}, i = j and ζ is a 5th root of unity.Their argument remains true for k-lines when char k = 2, 5. Let σ F be the section of E = Sym 5 S * → Gr(2, 5) defined by F .So σ −1 F (0) is the union of 50 irreducible components which we denote by W i for i = 1, . . ., 50.Furthermore, Albano and Katz study the lines on X which are limits of the 2875 lines on a family of threefolds with X 0 = X and find the following.Proposition 6.1 (Proposition 2.2 + 2.4 in [AK91]).For a general deformation X t there are exactly 10 complex lines in each component W i that deform with monodromy 2 in direction t.Proposition 6.2 (Proposition 2.3 + 2.4 in [AK91]).The lines that lie in the intersection of two components W i ∩ W j , i = j deform with monodromy 5 in direction t for a general deformation X t .This makes in total 2 • 10 • 50 + 375 • 5 = 2875 complex lines as expected (see e.g.[EH16]).Let We will show that their computations work over fields of characteristic not equal to 2 or 5 in Proposition 6.6 and Proposition 6.3 and thus we have found all the lines on X t → Spec k((t)).
To compute the dynamic Euler number of E = Sym 5 S * , we compute the sum of local indices at the lines on the base change X t ⊗ k((t)) = {F t = 0} ⊂ P 4 k((t)) , which we expect to contain 2875 lines defined over the algebraic closure of k((t)).The base change l t ⊗ k((t)) of one of the 2875 lines described in Proposition 6.6 and Proposition 6.3 lies on X t ⊗ k((t)).So we have found all the lines on X t ⊗ k((t)).By abuse of notation, we denote X t ⊗ k((t)) by X t and l t ⊗ k((t)) by l t .

Multiplicity 5 lines
Let l 0 be a line that lies in the intersection of two components W i ∩ W j with i = j.After base change we can assume that l 0 = {(u : −u : v : −v : 0)} ⊂ P 5 .As in (see [AK91, Proposition 2.4]) we introduce Z/5 monodromy: Since we expect l 0 to deform to order 5, we expect a deformation l t of l 0 to be a Spec L[[t 1/5 ]] where L is algebraic over k.In particular, we can replace t 1 5 by t and consider the deformation F t = F + t 5 G + t 10 H + . . . .We will show that l 0 deforms when char k = 2, 5 and there are 5 lines l t on the family X t over Spec(k[[t]]) with l t | t=0 = l 0 .over a field k with char k = 2, 5.
In the proof we computed the following coefficients of l t .
We now want to compute the contribution of all lines from Proposition 6.3 to (11).Recall that σ −1 F (0) is the union of 50 irreducible components and note that the union of lines in the intersection of two irreducible components are the lines in i,j,m,n where the union is over all pairwise different i, j, m, n ∈ {0, 1, 2, 3, 4}.

Using the local analytic structure
We want to present a different approach to finding the contribution of the multiplicity 5 lines.Clemens and Kley find that the local analytic structure at the crossings (that is at the intersection of 2 components) is That means that the local ring of σ −1 F (0) at a multiplicity 5 line is isomorphic to Observe that (f 1 , f 2 , f 5 , f 6 ) is a regular sequence, and setting x = 0, x = 0, y = 0 and y = 0 in f 3 and f 4 yields 10z 3 z 2 and 10z 2 z 3 .Dividing both polynomials by 10, we get Clemens and Kley's local structure.
So we get the same contribution as in (18).However, this does not reprove what has been done above.
To find the local A 1 -degree it does not suffice to remember the isomorphism class of the local ring.We need a presentation.That means, the order of the polynomials generating the ideal and the coefficients must not be forgotten.It works in this case because the product of the coefficients of the highlighted terms in (12) is a square.

Multiplicity 2 lines
Let l 0 be one of the lines described in Proposition 6.1 on one of the components W of σ −1 F (0).Let L 0 be the field of definition of l 0 .We introduce Z/2 monodromy and consider the deformation F t = F +t 2 G+t 4 H +. . . .Since l 0 does not lie in one of the intersections W ∩ W i (with W i = W ), we can assume that l 0 = {(u : −u : v : av : bv)} ⊂ P 5 with 1 + a 5 + b 5 = 0 and ab = 0.
In the proof we have calculated that x 1 = x 1 = 0 a 4 y 1 + b 4 z 1 = 0, a 4 y 1 + b 4 z 1 = 0 (w 2 +abd) .Then the contribution of the deformations of the 10 double lines on W to (11) is Tr E((t))/k((t)) (Tr L((t))/E((t)) ( Jf (l t ) )) where Jf (l t ) is again the determinant of the jacobian of f evaluated at the l t .
Again Jf (l t ) is determined by the lowest term of Jf (l t ), that is the lowest term of the determinant of the matrix  where the first sum runs over the lines that deform with multiplicity 2 and the second over the lines that deform with multiplicity 5 for a general deformation F t of the Fermat.
The image of e(E, σ) in CH d−r (Y ) under the inclusion CH d−r (σ −1 (0)) → CH d−r (Y ) is independent of the section σ and called the Euler class e(E) of π : E → Y .
the intersection product splits up as a sum of cycles supported on the distinguished varieties.If r = d and σ intersects s 0 transversally, then σ −1 (0) consists of the isolated zeros of σ which are the distinguished varieties of the intersection product (8).That means, e(E, σ) is supported on the isolated zeros of σ.When π : E → Y is also relatively oriented and Y is smooth and proper over an arbitrary field k (and still r = d), we have seen that the A 1 -Euler number e A 1 (E) is equal to the sum of local indices at the isolated zeros.In other words, the A 1 -Euler number e A 1 (E) is 'supported' on the zeros of a section with only isolated zeros, that is on the distinguished varieties.Oriented Chow groups CHi (Y, L) were introduced by Barge and Morel [BM00] and further studied by Fasel [Fas08].They are an 'oriented version' of Chow groups which can be defined for a (smooth) scheme Y over any field k.Here, L → Y is a line bundle and defines a 'twist' of the oriented Chow group.Levine defines an Euler class with support e Z (E, σ) ∈ CH d Z (Y, (det E) −1 ) in [Lev20, p.2191] which is supported on a closed subset Z ⊂ Y .Let L be a 1-dimensional k-vector space and denote by GW(k, L) the Grothendieck group of isometry classes of finite rank non-degenerate symmetric bilinear forms V × V → L. Then for a closed point x ∈ Y it holds that CH d x (Y, (det E) −1 ) = GW(k(x), (det E ⊗ det m x /m 2 0 and Y 4 = X 2 on P 4 .In the new coordinatesF = Y 5 3 + (Y 0 − Y 3 ) 5 + Y 5 4 + (Y 1 + aY 4 ) 5 + (Y 2 + bY 4 ) 5 .

)
Remark 6.7.As in the proof let Q(a, b) := g 3 (0) which is a degree 2 polynomial in a and b.The condition that l 0 deforms is d = 0 and Q(a, b) = 0 in [AK91, Proposition 2.2].Let E = k[a,b] (1+a 5 +b 5 ,Q(a,b)) .By the proof of Proposition 6.3 and Remark 6.7 the closed subscheme of the 10 multiplicity 2 lines on a component W that deform, has coordinate ring E = k[a,b] (a 5 +b 5 +1,Q(a,b)) .Let L := E[w]