Euler Characteristics of Crepant Resolutions of Weierstrass Models

Abstract

Based on an identity of Jacobi, we prove a simple formula that computes the pushforward of analytic functions of the exceptional divisor of a blowup of a projective variety along a smooth complete intersection with normal crossing. We use this pushforward formula to derive generating functions for Euler characteristics of crepant resolutions of singular Weierstrass models given by Tate’s algorithm. Since the Euler characteristic depends only on the sequence of blowups and not on the Kodaira fiber itself, several distinct Tate models have the same Euler characteristic. In the case of elliptic Calabi–Yau threefolds, using the Shioda–Tate–Wazir theorem, we also compute the Hodge numbers. For elliptically fibered Calabi–Yau fourfolds, our results also prove a conjecture of Blumenhagen, Grimm, Jurke, and Weigand based on F-theory/heterotic string duality.

Appendices

Jacobi’s Partial Fraction Identity

In this section, we prove a formula of Jacobi and exploit the theorem to give a simple proof of a formula of Louck and Biedenharn [53, Appendix A, p. 2400] by demonstrating its equivalence with the following theorem of Jacobi.

Theorem A.1

(Jacobi, [44, Section III.17, pp. 29–30]). Let $a_i$ ($i=1, \ldots , d$) be d distinct elements of an integral domain. Then

$$\begin{aligned} \prod _{i=1}^d \frac{1}{x-a_i}= \sum _{i=1}^d \frac{1}{x-a_i} \prod ^d_{\begin{array}{c} j=1\\ j\ne i \end{array}} \frac{1}{a_i-a_j}. \end{aligned}$$

(A.1)

Proof

Let

$$\begin{aligned} F(x)=\prod _{i=1}^d \frac{1}{x-a_i}, \end{aligned}$$

(A.2)

where $a_i\ne a_j$ for $i\ne j$. We would like to find the partial fraction expansion of F(x). That is, we would like to find coefficients $A_i$ ($i=1,\ldots , d$) such that

$$\begin{aligned} F(x)= \sum _{i=1}^d \frac{A_i}{x-a_i}. \end{aligned}$$

(A.3)

We determine $A_i$ by the method of residues. Multiplying (A.3) by $(x-a_i)$, simplifying, and evaluating at $x=a_i$ gives

$$\begin{aligned}\left. (x-a_j) F(x)\right| _{x=a_j}=A_j.\end{aligned}$$

Applying the above formula to (A.2), we get $A_j=\prod _{i\ne j} \frac{1}{a_i-a_j}$, which is the identity of Jacobi:

$$\begin{aligned} \prod _{i=1}^d \frac{1}{x-a_i}= \sum _{i=1}^d \frac{1}{x-a_i}\prod ^d_{\begin{array}{c} j=1\\ j\ne i \end{array}} \frac{1}{a_i-a_j}. \end{aligned}$$

(A.4)

$\square$

Theorem A.2

(Jacobi, Louck–Biedenharn, Cornelius). Let $h_r(x_1, \ldots , x_d)$ be the homogeneous complete symmetric polynomial of degree r in d variables of an integral domain. Then,

$$\begin{aligned} h_{r}(x_1, \ldots , x_d)= \sum _{\ell =1}^d x_\ell ^{r+d-1} \prod ^d_{\begin{array}{c} m=1\\ m\ne \ell \end{array}} \frac{1}{ x_\ell -x_m }. \end{aligned}$$

This theorem was proven by Louck-Biedenharn [53, Appendix A, p. 2400] and Cornelius [14]. We present a new and much simpler proof below by showing that the theorem is simply a reformulation of Jacobi’s identity (Theorem A.1).

Proof

Substituting $x\rightarrow 1/t$ in Eq. (A.1) gives:

$$\begin{aligned}&\prod _{i=1}^d \frac{t}{1-a_i t}= \sum _{i=1}^d \frac{t}{1-a_i t}\prod ^d_{\begin{array}{c} j=1\\ j\ne i \end{array}} \frac{1}{a_i-a_j}. \end{aligned}$$

Expanding $1/(1-a_i t)$ in both side of the equation gives

$$\begin{aligned}&t^{d}\sum _{r=0}^\infty h_r(a_1, \ldots , a_d) t^r= t\sum _{i=1}^d \sum _{k=0}^\infty a_i^k t^k \prod ^d_{\begin{array}{c} j=1\\ j\ne i \end{array}} \frac{1}{a_i-a_j}\\&\sum _{r=0}^\infty h_r(a_1, \ldots , a_d) t^{r+d-1}= \sum _{k=0}^\infty \Big ( \sum _{i=1}^d a_i^k \prod ^d_{\begin{array}{c} j=1\\ j\ne i \end{array}} \frac{1}{a_i-a_j}\Big ) t^k. \end{aligned}$$

Comparing terms of the same degree in t, we get the final expression of Lemma 1.10:

$$\begin{aligned} h_{r} (a_1,\ldots , a_d)=\sum _{i=1}^d a_i^{r+d-1} \prod ^d_{\begin{array}{c} j=1\\ j\ne i \end{array}} \frac{1}{a_i-a_j}. \end{aligned}$$

$\square$

The Euler Characteristic as the Degree of the Top Chern Class

The purpose of this section is to explain from different points of view why the Euler characteristic is the degree of the top Chern class. Traditionally, this statement is seen as a generalization of the Poincaré–Hopf theorem that asserts that the total degree of a vector field defined on a smooth manifold M is the Euler characteristic of M. This statement can also be seen as a generalization of the Gauss–Bonnet–Chern Theorem (which is itself is a consequence of Poincaré–Hopf theorem). Here we will review three different approaches. The first one relies on Leftschetz fixed point theorem. The second one uses he Poincaré–Hopf theorem using the interpretation of Chern classes as related to the class of some degenerated loci as discussed in Chapter 3 of Fulton. The third one is an application of the Hirzebruch–Riemann–Roch theorem and the Hodge decomposition theorem.

Let M be a smooth compact manifold. The kth Betti number of M is by definition the dimension of the cohomology group $H^k(M,\mathbb {Q})$. The Euler characteristic of M is denoted by $\chi (X)$ and is defined as the following alternative sum of Betti numbers of M:

$$\begin{aligned} \chi (M):=\sum _{k=0}^{\dim M} (-1)^k b_k, \quad b_k:= \dim H^i(M,\mathbb {Q}). \end{aligned}$$

1.1 Lefschetz fixed point theorem and the Euler characteristic as an intersection number

Theorem B.1

(Lefschetz fixed point theorem). Let M be a compact smooth manifold of dimension m and $f:M\longrightarrow M$ a continuous map. We define the Lefschetz number of f as

$$\begin{aligned} L(f):=\sum _{k=0}^m (-1)^k\ {\mathrm {tr}} \Big (f^*|H^k(M,\mathbb {Q})\Big ), \quad f^*: H^k(M,\mathbb {Q})\longrightarrow H^k(M,\mathbb {Q}). \end{aligned}$$

Then L(f) is equal to the intersection number of the graph $\Gamma _f$ of f and the diagonal $\Delta$ in $M\times M$

$$\begin{aligned} L(f)=\int _{M\times M} \Gamma _f \cdot \ \Delta . \end{aligned}$$

Thus, the Leftschetz number L(f) is the number of fixed points of f counted with multiplicities.

Corollary

Let M be a compact smooth manifold and $\Delta$ be the diagonal of $M\times M$, then the Euler characteristic of M, $\chi (M)=\sum _i (-1)^k \dim H^i(M, \mathbb {Q})$, is equal to the self-intersection of $\Delta$ in $M\times M$:

$$\begin{aligned} \chi (M)=\int _{M\times M} \Delta \cdot \Delta . \end{aligned}$$

Proof

Consider the special case of Lefschetz theorem for which f is the identify map on M. Then, the Leftschetz number reduces to the Euler characteristic $\chi (M)$ as the trace ${\mathrm {tr}} \Big (f^*|H^k(M,\mathbb {Q})\Big )$ becomes the kth Betti number $b^k$ of M and the intersection number $\int _{M\times M} \Gamma _{f} \cdot \ \Delta$ becomes the self-intersection of the diagonal $\Delta$ in $M\times M$. $\square$

Theorem B.2

(Self-intersection formula, see [37, Corollary 6.3, pp. 102–103]). Let $i: Z\rightarrow X$ be a regular imbedding of codimension d and normal bundle N. Then for any $\alpha \in A_*(Z)$ we have the self-intersection formula

$$\begin{aligned} i^* i_* (\alpha )=c_d(N)\cap \alpha . \end{aligned}$$

Theorem B.3

If X is a nonsingular complete algebraic variety, then the Euler characteristic of X is equal to the degree of the total homological Chern class of X:

$$\begin{aligned} \chi (X)=\int c(X), \quad c(X):=c(TX)\cap [X]. \end{aligned}$$

Proof

The theorem follows from the previous corollary expressing the Euler characteristic $\chi (X)$ as the self-intersection of the diagonal $\Delta$ in $X\times X$, followed by the self-intersection formula expressing $\Delta \cdot \Delta$ as the class $c_{\dim X} (N_\Delta {X\times X})\cap [\Delta ]$. Since the normal bundle of $\Delta$ in $X\times X$ is isomorphic to the tangent bundle of X (see for example [11, Lemma 11.23, p. 127]), it follows that [37, Example 8.1.12, p. 136], the self-intersection of the diagonal $\Delta$ in $X\times X$ is $\int c_{\dim X} (TX)\cap [X]=\int c(TX)\cap [X]$:

$$\begin{aligned} \begin{aligned} \chi (X)&=\int _{X\times X} \Delta \cdot {\Delta }=\int c(N_\Delta {X\times X})\cap [\Delta ]=\int c(TX)\cap [X]. \end{aligned} \end{aligned}$$

$\square$

1.2 Poincaré-Hopf theorem and the Euler characteristic

Theorem B.4

(Poincaré-Hopf). Let M be a smooth compact manifold without boundary and v be a vector field with isolated zeros. Then the sum of the local indices at the zeros of v is equal to the Euler characteristic of M.

Remark B.5

This theorem can be generalized to manifolds with boundaries by requiring v to point outward. Poincaré proved a two dimensional version of this theorem in 1885. The general version was proven by Hopf in 1926.

Theorem B.6

([37, Example 3.2.16, p. 61]). Let E be a vector bundle of rank r on a smooth variety X, let s be a section of E, and Z the zero-scheme of s. If X is purely n-dimensional and s is a regular section, then Z is purely $(n-r)$-dimensional, and

$$\begin{aligned} {[Z]}=c_r(E)\cap [X]. \end{aligned}$$

In particular, if E is the tangent bundle TX of X, then r (i.e. the rank of E) is the dimension of X, and the section s of E is just a vector field. The zero-scheme Z is a 0-cycle that is the sum of the isolated singularities of s counted with multiplicities. Hence, the degree of the top Chern class of TX gives the index of the vector field s, which is the Euler characteristic of M by the Poincaré–Hopf theorem. Since the degree of c(X) is exactly the degree of $c_{r}(TX)\cap [X]$, we retrieve Theorem B.3:

$$\begin{aligned} \chi (X)=\int c(X). \end{aligned}$$

1.3 Hirzebruch–Riemann–Roch theorem and the Euler characteristic

In this sub-section, using the Hirzebruch–Riemann–Roch theorem and the Hodge decomposition theorem, we prove that the Euler characteristic of a nonsingular projective variety is the degree of its homological total Chern class. We follow Fulton ([37, Example 18.3.7, p. 362] and [37, Example 3.2.5, p. 57]) as presented by Rössler [67]. We denote the Todd class, the Chern character, and the dual of a vector bundle E by ${\mathrm {td}}(E)$ and ${\mathrm {ch}}(E)$, and $E^\vee$ respectively.

Let X be a projective variety of dimension d and V a coherent sheaf defined over X. We denote by $H^q(X,V)$ the q-th cohomology group of X with coefficients in the sheaf of germs of local sections of V. The cohomology groups $H^q(X,V)$ vanish for $q>d$ and are all finite dimensional for $0 \le q\le d$. The Euler characteristic of V in X is by definition the finite number

$$\begin{aligned} \chi (X,V):=\sum _{q=0}^d (-1)^q \dim H^q(X, V). \end{aligned}$$

The Hirzebruch–Riemann–Roch theorem provides an expression for $\chi (X,V)$ in terms of characteristic classes of TX and V realizing a conjecture of Serre in a letter to Kodaira and Spencer.

Theorem B.7

(Hirzebruch–Riemann–Roch). Let V be a coherent sheaf over a nonsingular variety X. Then

$$\begin{aligned} \chi (X,V)=\int _X{\mathrm {ch}}(V) {\mathrm {td}}(TX). \end{aligned}$$

We will also need the following lemma relating the Todd class and the Chern character. This lemma is instrumental in the proof of the Hirzebruch–Riemann–Roch theorem of Borel and Serre [10, Lemma 18, p. 128], and is also discussed by Fulton in [37, Example 3.2.5, p. 57].

Lemma B.8

(Hirzebruch [41, Theorem 10.1.1, p. 92])). Let E be a vector bundle of rank r. Then

$$\begin{aligned} {\mathrm {ch}}\Big (\sum _{q=0}^r (-1)^q {\bigwedge }^{q} E^\vee \Big ) {\mathrm {td}}(E)=c_r(E). \end{aligned}$$

Proof

By the splitting principal, we can always formally factorize the total Chern class of E as $c(E)=\prod _i (1+ a_i )$, where $a_i$ are the Chern roots of E. Then by definition

$$\begin{aligned} {\mathrm {ch}}(E):=\sum _{i=1}^r e^{a_i}, \quad {\mathrm {td}}(E):=\prod _i \frac{a_i}{(1-e^{-a_i})}. \end{aligned}$$

We have the classical relations (see [41, Theorem 4.4.3, p. 64] or [37, Remark 3.2.3, pp. 54–56])

$$\begin{aligned} c(E^\vee )=\prod _i (1-a_i), \quad c({\bigwedge }^q E)= \prod _{1\le i_1< \cdots < i_q\le r} \Big (1+a_{i_1} +\cdots +a_{i_q}\Big ). \end{aligned}$$

Hence

$$\begin{aligned} \quad {\mathrm {ch}}\Big ({ \bigwedge }^q E^\vee \Big )= \sum _{1\le i_1< \cdots < i_q\le r} e^{-(a_{i_1} +\cdots +a_{i_q})}. \end{aligned}$$

Thus by the additive properties of the Chern character and the definition of the Todd class:

$$\begin{aligned} {\mathrm {ch}}\Big (\sum _{q=0}^r (-1)^q {\bigwedge }^{q} E^\vee \Big )&= \sum _{q=0}^r (-1)^q {\mathrm {ch}}\Big ({\bigwedge }^q E^\vee \Big )=\prod _{i=1}^r (1- e^{-a_{i}})\\&=(a_1\ldots a_r) \prod _{i=1}^r \frac{(1- e^{-a_{i}})}{a_i}=c_r(E) {\mathrm {td}}^{-1}(E). \end{aligned}$$

$\square$

Theorem B.9

Let X be a nonsingular complete projective variety defined over the complex numbers. Then the Euler characteristic

$$\begin{aligned} \chi (X)=\int c(X). \end{aligned}$$

Proof

For X a nonsingular variety of dimension d, we apply Lemma B.8 to the tangent bundle $E=TX$ and we note that $E^\vee =TX^\vee :=\Omega _X$, where $\Omega _X$ is the sheaf of differentials of X, and by definition, the sheaf of differential p-forms is $\bigwedge ^q \Omega _X :=\Omega _X^q$. Hence, we get

$$\begin{aligned} {\mathrm {ch}}\Big (\sum _{q=0}^d (-1)^q \Omega _X^q \Big ) {\mathrm {td}}(TX)=c_r(TX). \end{aligned}$$

We rewrite the left hand side of the previous equation as follows

$$\begin{aligned} \int _X{\mathrm {ch}}\Big (\sum _{q=0}^d (-1)^q \Omega _X^q\Big ) {\mathrm {td}}(TX)= & {} \sum _{q=0}^d (-1)^q \int _X{\mathrm {ch}} ( \Omega ^q_X) {\mathrm {td}}(TX)\\= & {} \sum _{q=0}^d (-1)^{q}\int _X \chi (X, \Omega _X^q )\\= & {} \sum _{q=0}^d \sum _{p=0}^d (-1)^{p+q} {\mathrm{dim}}\ H^p (X, \Omega _X^q)\\= & {} \sum _{k=0}^d (-1)^k \sum _{p+q=k} {\mathrm{dim}}\ H^p (X, \Omega _X^q)\\= & {} \sum _{k=0}^d (-1)^k b_k\\= & {} \chi (X). \end{aligned}$$

The first equality is a direct consequence of the additive property of the Chern character, the second equality is due to the Hirzebruch–Riemann-Roch theorem applied to $\Omega _X^q$, the third equality follows from the definition of the Euler characteristic of a sheaf, and the fifth equality is a direct application of the Hodge decomposition theorem $\Omega ^k=\bigoplus _{p+q=k} \Omega ^{p,q}$ and Dolbeault’s theorem, which asserts that the Dolbeault cohomology is isomorphic to the sheaf cohomology of the sheaf of differential forms: $H^{p,q}(X)\cong H^p (X, \Omega _X^q).$ In particular, $h^{p,q}(X)={\mathrm{dim}}\ H^p (X, \Omega _X^q)$ are the Hodge numbers of X. The last equality is by the definition of the Euler characteristic. Hence, since $\int c(X)=\int c(TX)\cap [X]=\int _X c_r (TX)$, we get

$$\begin{aligned} \int c(X)=\chi (X). \end{aligned}$$

$\square$

Basic Notions

The local ring of a subvariety S of X is denoted $\mathscr {O}_{X,S}$, its maximal ideal is $\mathscr {M}_{X,S}$ and the quotient field is the residue field $\kappa (S)=\mathscr {O}_{X,S}/\mathscr {M}_{X,S}$. The local ring $\mathscr {O}_{X,S}$ is the stalk of the structure sheaf of X at the generic point $\eta _S$ of S and $\kappa (S)$ is the function field of S. If S is a divisor, $\mathscr {O}_{X,S}$ is a one dimensional local domain. In case X is nonsingular along S, $\mathscr {O}_{X,S}$ is a discrete valuation ring and the order of vanishing is given by the usual valuation.

1.1 Fiber types, dual graphs, Kodaira symbols

Definition C.1

(Algebraic cycle). An algebraic cycle of a Noetherian scheme X is a finite formal sum $\sum _i N_i V_i$ of subvarieties $V_i$ with integer coefficients $N_i$. If all the subvarieties $V_i$ have the same dimension d, the cycle is called a d-cycle. The free group generated by subvarieties of dimension d is denoted $Z_d(X)$. The group of all cycles, denoted $Z(X)=\bigoplus _d Z_d(X)$, is the free group generated by subvarieties of X.

Definition C.2

(Degree of a zero-cycle [37, Chapter 1, Definition 1.4, p. 13]). Let X be a complete scheme. The degree of a zero-cycle $\sum N_i p_i$ of X is

$$\begin{aligned} {\mathrm {deg}} (\sum _i N_i p_i)=\sum _i N_i [\kappa (p_i): k],\end{aligned}$$

where $[\kappa (p_i): k]$ is the degree of the field extension $\kappa (p_i)\rightarrow k$.

Let $\Theta$ be an algebraic one-cycle with irreducible decomposition $\Theta =\sum _i m_i \Theta _i$. We denote by $\Theta _i \cdot \Theta _j$ the zero-cycle defined by the intersection of $\Theta _i$ and $\Theta _j$ for $i\ne j$.

Definition C.3

(n-points, tree). A n-point of an algebraic one-cycle $\Theta$ is a point in $\bigcup _i \Theta _i$, which belongs to exactly n distinct irreducible components $\Theta _i$. An algebraic one-cycle $\Theta$ is said to be a tree if it does not have n-points for $n>2$. Two curves intersect transversally if their intersection consists of isolated reduced closed points.

**Following Kodaira [47, 48], we introduce the following definition:

Definition C.4

(Fiber type). By the type of an algebraic one-cycle $\Theta \in Z_1(X)$ with irreducible decomposition $\Theta =\sum _i m_i \Theta _i$, we mean the isomorphism class of each irreducible curve $\Theta _i$, together with the topological structure of the reduced polyhedron $\sum \Theta _{i}$ (that is the collection of zero-cycles $\Theta _i\cdot \Theta _j$ ($i\ne j$)), and the homology class of $\Theta =\sum _i m_i \Theta _{i}$ in the Chow group $A_1(X)$.

Example C.5

For instance, $\Theta _1\cdot \Theta _2= 2p_1 +3 p_2$ indicates that the two curves $\Theta _1$ and $\Theta _2$ meet at two points $p_1$ and $p_2$ with respective intersection multiplicity 2 and 3.

Definition C.6

(Dual graph). To an algebraic one-cycle $\Theta$ with irreducible decomposition $\Theta =\sum _i m_i \Theta _{i}$, we associate a weighted graph (called the dual graph of $\Theta$) such that:

• The vertices are the irreducible components of the fiber.

• The weight of a vertex corresponding to the irreducible component $\Theta _i$ is its multiplicity $m_i$. When the multiplicity is one, it can be omitted.

• The vertices corresponding to the irreducible components $\Theta _i$ and $\Theta _j$ ($i\ne j$) are connected by $\hat{\Theta }_{i,j}={\mathrm {deg}}( \Theta _i\cdot \Theta _j)$ edges.

Definition C.7

(Kodaira symbols, See [47, 48, Theorem 6.3]). Kodaira has introduced the following symbols characterizing the type of one-cycles appearing in the study of minimal elliptic surfaces. See Table 4 for a visualization of these fibers.

• 1. Type $\hbox {I}_0$: a smooth curve of genus 1.

• 2. Type $\hbox {I}_1$: an irreducible nodal rational curve.

• 3. Type II: an irreducible cuspidal rational curve.

• 4. Type $\hbox {I}_2$: $\Theta =\Theta _1+\Theta _2$ and $\Theta _1\cdot \Theta _2=p_1+p_2$: two smooth rational curves intersecting transversally at two distinct points $p_1$ and $p_2$. The dual graph of $\hbox {I}_2$ is $\widetilde{A}_1$.

• 5. Type III: $\Theta =\Theta _1+\Theta _2$ and $\Theta _1\cdot \Theta _2=2p$: two smooth rational curves intersecting at a double point. Its dual graph is $\widetilde{A}_1$.

• 6. Type IV: $\Theta =\Theta _1+\Theta _2+\Theta _3$ and $\Theta _1\cdot \Theta _2=\Theta _1\cdot \Theta _3=\Theta _2\cdot \Theta _3=p$: a 3-star composed of smooth rational curves. Its dual graph is $\widetilde{A}_2$.

• 7. Type $\hbox {I}_n$$(n\ge 3)$: $\Theta =\Theta _0+\cdots \Theta _n$ with $\Theta _i\cdot \Theta _{i+1} =p_i$$i=0,\ldots , {n-1}$ and $\Theta _n \cdot \Theta _0= p_n$. Its dual graph is the affine Dynkin diagram $\widetilde{A}_{n-1}$.

• 8. Type $\hbox {I}^*_n$$(n\ge 0)$: $\Theta =\Theta _0+\Theta _1 + 2 \Theta _{2}+\cdots +2\Theta _{n+2}+\Theta _{n+3}+\Theta _{n+4}$, with $\Theta _{i}\cdot \Theta _{i+1}=p_{i}$$(i=1,\ldots , n+2)$, $\Theta _0\cdot \Theta _2=p_0$, $\Theta _{n+4}\cdot \Theta _{n+2}=p_{n+4}$. The dual graph the affine Dynkin diagram $\widetilde{D}_{4+n}$.

• 9. Type $\hbox {IV}^*$: $\Theta =\Theta _0+\Theta _1 +2 \Theta _2+2\Theta _3 +3\Theta _4 +2\Theta _5 +\Theta _6$ with $\Theta _{i}\cdot \Theta _{i+1}=p_i$ ($i=3,\ldots ,6$), $\Theta _1\cdot \Theta _3=p_1$, $\Theta _0\cdot \Theta _2=p_0$, $\Theta _2\cdot \Theta _4=p_2$. The dual graph is the affine Dynkin diagram $\widetilde{E}_{6}$.

• 10. Type $\hbox {III}^*$: $\Theta =\Theta _0+2\Theta _1 +2 \Theta _2+3\Theta _3 +4\Theta _4 +3\Theta _5 +2\Theta _6 + \Theta _7$ with $\Theta _{i}\cdot \Theta _{i+1}=p_i$ ($i=3,\ldots ,6$), $\Theta _1\cdot \Theta _3=p_1$, $\Theta _0\cdot \Theta _1=p_0$, $\Theta _2\cdot \Theta _4=p_2$. The dual graph is the affine Dynkin diagram $\widetilde{E}_{7}$.

• 11. Type $\hbox {II}^*$: $\Theta =2\Theta _1 +3 \Theta _2+4\Theta _3 +6\Theta _4 +5\Theta _5 +4\Theta _6 + 3\Theta _7 + 2\Theta _8 + \Theta _0,$ with $\Theta _{i}\cdot \Theta _{i+1}=p_i$ ($i=3,\ldots ,7$), $\Theta _1\cdot \Theta _3=p_1$, $\Theta _8\cdot \Theta _0=p_8$, and $\Theta _2\cdot \Theta _4=p_2$. The dual graph the affine Dynkin diagram $\widetilde{E}_{8}$.

1.2 Elliptic fibrations, generic versus geometric fibers

Definition C.8

(Elliptic fibrations). A surjective proper morphism $\varphi :Y\longrightarrow B$ between two algebraic varieties Y and B is called an elliptic fibration if the generic fiber of $\varphi$ is a smooth projective curve of genus one and $\varphi$ has a rational section. When B is a curve, Y is called an elliptic surface. When B is a surface, Y is said to be an elliptic threefold. In general, if B has dimension $n-1$, Y is called an elliptic n-fold.

The locus of singular fibers of $\varphi$ is called the discriminant locus of $\varphi$ and is denoted $\Delta (\varphi )$ or simply $\Delta$ when the context is clear. If the base B is smooth, the discriminant locus is a divisor [20]. The singular fibers of a minimal elliptic surface have been classified by Kodaira and Néron. The dual graphs of these geometric fibers are affine Dynkin diagrams. We denote these singular fibers by their Kodaira symbols as described in Definition C.7 and presented in Table 4.

The language of schemes streamlines many notions in the study of fibrations. We review some basic definitions.

Definition C.9

(Fiber over a point). Let $\varphi : Y\longrightarrow B$ be a morphism of schemes. For any $p\in B$, the fiber over p is denoted $Y_p$ and defined using a fibral product⁵ as

$$\begin{aligned}Y_p=Y\times _B {\mathrm{Spec}}\ \kappa (p).\end{aligned}$$

The first projection $Y_p\longrightarrow Y$ induces an homeomorphism from $Y_p$ onto $f^{-1}(p)$ [52, §3.1 Proposition 1.16] . The second projection gives $Y_p$ the structure of a scheme over the residue field $\kappa (p)$.

If p is not a closed point,⁶ the residue field $\kappa (p)$ is not necessarily algebraically closed. Certain components of $Y_p$ could be $\kappa (p)$-irreducible (i.e. irreducible when defined over $\kappa (p)$) while they become reducible after an appropriate field extension. An irreducible scheme over a field k is said to be geometrically irreducible when it stays irreducible after any field extension. The most refined description of the fiber $Y_p$ is always the one corresponding to the algebraic closure $\overline{\kappa (p)}$ of $\kappa (p)$. This motivates the following definition.

Definition C.10

The geometric fiber over p is the fiber $Y_p\times _{\kappa (p)} \overline{\kappa (p)}$, the fiber $Y_p$ after the base change induced by the field extension $\kappa (p)\rightarrow \overline{\kappa (p)}$ to the algebraic closure of $\kappa (p)$.

By construction, a geometric fiber is always composed of geometrically irreducible components.

Definition C.11

We say that the type of a fiber $Y_p$ is geometric if it does not change after a field extension.

Remark C.12

To emphasize the difference between the fiber $Y_p$ and its geometric fiber, we will refer to the fiber $Y_p$ (defined with respect to the residue field $\kappa (p)$) as the arithmetic fiber.

For an elliptic n-fold, the Kodaira fibers are also the geometric generic fibers of the irreducible components of the reduced discriminant locus. While the dual graph of a Kodaira fiber is an affine Dynkin diagram of type $\widetilde{A}_k$, $\widetilde{D}_{4+k}$, $\widetilde{E}_6$, $\widetilde{E}_7$, or $\widetilde{E}_8$, the dual graph of the generic (arithmetic) fiber itself can also be a twisted Dynkin diagram of type $\widetilde{B}_{3+k}^t$, $\widetilde{C}_{2+k}^t$, $\widetilde{G}_2^t$, or $\widetilde{F}_4^t$. This is reviewed in Tables 2 and 3. These dual graphs are not geometric in the sense that after an appropriate base change they become $\widetilde{D}_{4+n}$, $\widetilde{A}_{2+2k}$ or $\widetilde{A}_{1+2k}$, and $\widetilde{E}_6$ respectively. The Kodaira fibers of the following type never need a field extension:

$$\begin{aligned} \hbox {I}_1, \hbox {II}, \hbox {III}, \hbox {III}^{*},~ \hbox {and} ~ \hbox {II}^*. \end{aligned}$$

The remaining Kodaira fibers (IV, $\hbox {I}_{n>1}$, $\hbox {I}_n^*$, and $\hbox {IV}^*$) can come from fibers $\hbox {Y}_p$ whose types are not geometric and require at least a field extension of degree 2 to describe a fiber with a geometric type. When the fiber $Y_p$ has a geometric type, the type of the fiber is said to be split. Otherwise, the type of $Y_p$ is said to be non-split. When that is the case we mark the fiber with an “ns” superscript: $\hbox {IV}^{\text {ns}}$, $\hbox {I}_n^{\text {ns}}$, $\hbox {I}_n^{*\text {ns}}$, $(n\ge 2)$ and $\hbox {IV}^{*\text {ns}}$. When a field extension is not needed, the fibers are marked with an “s” superscript (“split”): $\hbox {IV}^{\text {s}}$, $\hbox {I}_n^{\text {s}}$, $\hbox {I}_n^{*\text {s}}$, $(n\ge 2)$ and $\hbox {IV}^{*\text {s}}$. The fiber of type $\hbox {I}_0^*$ can be split, semi-split, or non-split if the Kodaira types require no field extension, a quadratic extension, or a cubic extension. The corresponding dual graphs are respectively $\widetilde{D}_4$, $\widetilde{B}_3^t$, and $\widetilde{G}_2^t$.

1.3 Weierstrass models and Deligne’s formulaire

We follow the notation of Deligne [16]. Let $\mathscr {L}$ be a line bundle over a quasi-projective variety B. We define the following projective bundle (of lines):

$$\begin{aligned} \pi : X_0=\mathbb {P}_B[\mathscr {O}_B\oplus \mathscr {L}^{\otimes 2}\oplus \mathscr {L}^{\otimes 3}]\longrightarrow B. \end{aligned}$$

(C.1)

The relative projective coordinates of $X_0$ over B are denoted [z : x : y], where z, x, and y are defined respectively by the natural injection of $\mathscr {O}_B$, $\mathscr {L}^{\otimes 2}$, and $\mathscr {L}^{\otimes 3}$ into $\mathscr {O}_B\oplus \mathscr {L}^{\otimes 2}\oplus \mathscr {L}^{\otimes 3}$. Hence, z is a section of $\mathscr {O}_{X_0}(1)$, x is a section of $\mathscr {O}_{X_0}(1)\otimes \pi ^*\mathscr {L}^{\otimes 2}$, and y is a section of $\mathscr {O}_{X_0}(1)\otimes \pi ^*\mathscr {L}^{\otimes 3}$.

Definition C.13

A Weierstrass model is an elliptic fibration $\varphi : Y\rightarrow B$ cut out by the zero locus of a section of the line bundle $\mathscr {O}(3)\otimes \pi ^*\mathscr {L}^{\otimes 6}$ in $X_0$.

The most general Weierstrass equation is written in the notation of Tate as [16] $F=0$ with

$$\begin{aligned} F=y^2z+ a_1 xy z + a_3 yz^2 -(x^3+ a_2 x^2 z + a_4 x z^2 + a_6 z^3), \end{aligned}$$

(C.2)

where $a_i$ is a section of $\pi ^*\mathscr {L}^{\otimes i}$. The line bundle $\mathscr {L}$ is called the fundamental line bundle of the Weierstrass model $\varphi :Y\rightarrow B$ and can be defined directly from the elliptic fibration Y as $\mathscr {L}=R^1 \varphi _*\mathscr {O}_Y$. Following Tate and Deligne, we introduce the following quantities [16]

$$\begin{aligned} {\left\{ \begin{array}{ll} b_2 &{}= a_1^2 + 4 a_2\\ b_4 &{}= a_1 a_3 + 2 a_4\\ b_6 &{}= a_3^2 + 4 a_6\\ b_8 &{}= a_1^2 a_6 - a_1 a_3 a_4 + 4 a_2 a_6 + a_2 a_3^2 - a_4^2\\ c_4 &{}= b_2^2 - 24 b_4\\ c_6 &{}= -b_2^3 + 36 b_2 b_4 - 216 b_6\\ \Delta &{}= -b_2^2 b_8 - 8 b_4^3 - 27 b_6^2 + 9 b_2 b_4 b_6\\ j&{} = {c_4^3}/{\Delta } \end{array}\right. }. \end{aligned}$$

(C.3)

These quantities satisfy the following two relations

$$\begin{aligned} 1728 \Delta =c_4^3-c_6^2, \quad 4b_8 = b_2 b_6 - b_4^2. \end{aligned}$$

(C.4)

The $b_i$ ($i=2,3,4,6)$ and $c_i$ ($i=4,6$) are sections of $\pi ^*\mathscr {L}^{\otimes i}$. The discriminant $\Delta$ is a section of $\pi ^*\mathscr {L}^{\otimes 12}$. Geometrically, the discriminant $\Delta$ is the locus of points over which the elliptic fiber is singular. The j-invariant characterizes a smooth elliptic curve up to isomorphism. If we complete the square in y in the Weierstrass equation, the equation becomes

$$\begin{aligned} zy^2 =x^3 +\tfrac{1}{4}b_2 x^2 z + \tfrac{1}{2} b_4 x z^2+ \tfrac{1}{4} b_6 z^3. \end{aligned}$$

(C.5)

In addition, if we complete the cube in x gives the short form of the Weierstrass equation, the equation becomes

$$\begin{aligned} zy^2 =x^3 -\tfrac{1}{48} c_4 x z^2 -\tfrac{1}{864} c_6 z^3. \end{aligned}$$

(C.6)

1.4 Tate’s algorithm

Let R be a complete discrete valuation ring with valuation v, uniformizing parameter s, and perfect residue field $\kappa =R/(s)$. We are interested in the case where $\kappa$ has characteristic zero. We recall that a discrete valuation ring has only three ideals, the zero ideal, the ring itself, and the principal ideal sR. We take the convention in which the ring itself is not a prime ideal. It follows that the scheme ${\mathrm {Spec}}(R)$ has only two points: the generic point (defined by the zero ideal) and the closed point (defined by the principal ideal sR).

Let E / R be an elliptic curve over R with Weierstrass equation

$$\begin{aligned} y^2+a_1 xy + a_3 y=x^3+a_2 x^2+ a_4 x + a_6, \quad a_i \in R. \end{aligned}$$

The generic fiber is a regular elliptic curve. After a resolution of singularities, we have a regular model $\mathscr {E}$ over R and the special fiber is the fiber over the closed point ${\mathrm {Spec}}\ R/ (s)$. Tate’s algorithm determines the type of the geometric fiber over the closed point of ${\mathrm {Spec}} (R)$ by manipulating the valuations of the coefficients and the discriminant, and the arithmetic properties of some auxiliary polynomials. The type of the geometric fiber is denoted by its Kodaira’s symbol (see Definition C.7). The special fiber becomes geometric after a quadratic or a cubic field extension $\kappa '/\kappa$. Keeping track of the field extension used gives a classification of the special fiber as a $\kappa$-scheme—this is what we call the arithmetic fiber. The information on the required field extension needed to have geometrically irreducible components is already carefully encoded in Tate’s original algorithm, as it is needed to compute the local index (denoted by c in Tate’s notation). In the language of Néron’s model, the local index c is the order of the component group; geometrically, the local index is the number of reduced components of the special fiber defined over $\kappa$. . Following Tate, we use the convenient notation

$$\begin{aligned} a_{i,j}= a_i s^{-j}. \end{aligned}$$

Tate’s algorithm consists of the following eleven steps (see [72, 69, §IV.9], [19, 23, 42]). For Step 7, we use the more refined description of Papadopoulos [42, Part III, p. 134] who also gives in [42, §1, p. 122] an exhaustive list of errata of Tate’s original paper [72]. Tate’s algorithm is discussed in F-theory in [8, 46]. Subtleties in Step 6 and the distinction between two $\hbox {G}_2$-models depending on $[\kappa ':\kappa ]$ are explained in [27]. We follow the presentation of [23]:

• Step 1 $v(\Delta )=0 \implies$$\hbox {I}_0$.

• Step 2. If $v(\Delta )\ge 1$, change coordinates so that $v(a_3)\ge 1$, $v(a_4)\ge 1$, and $v(a_6)\ge 1$.

  If $v(b_2)=0$, the type is $\hbox {I}_{v(\Delta )}$. To have a fiber with geometric irreducible components, it is enough to work in the splitting field $\kappa '$ of the following polynomial of $\kappa [T]$:

  $$\begin{aligned} T^2+a_1 T-a_2. \end{aligned}$$

  The discriminant of this quadric is $b_2$. If $b_2$ is a square in $\kappa$, then $\kappa '=\kappa$, otherwise $\kappa '\ne \kappa$: (a) $\kappa '=\kappa \implies$$\hbox {I}_n^\text {s}$    (b) $\kappa '\ne \kappa \implies$$\hbox {I}_n^{\text {ns}}$

• Step 3. $v(b_2)\ge 1$, $v(a_3)\ge 1$, $v(a_4)\ge 1$, and $v(a_6)= 1$$\implies$ II.

• Step 4. $v(b_2)\ge 1$, $v(a_3)\ge 1$, $v(a_4)= 1$, and $v(a_6)\ge 2$$\implies$ III.

• Step 5. $v(b_2)\ge 1$, $v(a_3)\ge 1$, $v(a_4)\ge 2$, $v(a_6)\ge 2$, and $v(b_6)=2$$\implies$ IV. The fiber has geometric irreducible components over the splitting field $\kappa '$ of the polynomial

  $$\begin{aligned} T^2+a_{3,1} T-a_{6,2}. \end{aligned}$$

  Its discriminant is $b_{6,2}$. If $b_{6,2}$ is a square in $\kappa$, then $\kappa '=\kappa$ otherwise $\kappa '\ne \kappa$.

  (a) $\kappa '=\kappa \implies$$\hbox {IV}^\text {s}$    (b) $\kappa '\ne \kappa \implies$$\hbox {IV}^{\text {ns}}$

• Step 6. $v(b_2)\ge 1$, $v(a_3)\ge 1$, $v(a_4)\ge 2$, $v(a_6)\ge 3$, $v(b_6)\ge 3$, $v(b_8)\ge 3$. Then make a change of coordinates such that $v(a_1)\ge 1$, $v(a_2)\ge 1$, $v(a_3)\ge 2$, $v(a_4)\ge 2$, and $v(a_6)\ge 3$. Let

  $$\begin{aligned} P(T)= T^3+ a_{2,1} T^2 + a_{4,2} T + a_{6,3}. \end{aligned}$$

  If P(T) is a separable polynomial in $\kappa$, that is if P(T) has three distinct roots in a field extension of $\kappa$, then the type is $\hbox {I}_0^*$. The geometric fiber is defined over the splitting field $\kappa '$ of P(T) in $\kappa$. The type of the special fiber before to go to the splitting field depends on the degree of the field extension $\kappa '\rightarrow \kappa$:

  • $[\kappa ':\kappa ]=6\ \text {or}\ 3\implies$$\hbox {I}_0^{*\text {ns}}$ with dual graph $\widetilde{\text {G}}_2^t$.

  • $[\kappa ':\kappa ]=2\quad \quad \implies$$\hbox {I}_0^{*\text {ss}}$ with dual graph $\widetilde{\text {B}}_3^t$.

  • $[\kappa ':\kappa ]=1\quad \quad \implies$$\hbox {I}_0^{*\text {s}}$ with dual graph $\widetilde{\text {D}}_4$

  where “ns”, “ss”, and “s” stand respectively for “non-split”, “semi-split”, and “split”. In the notation of Liu, these fibers are respectively $\hbox {I}^*_{0,3}$, $\hbox {I}^*_{0,2}$, and $\hbox {I}^*_{0}$. The Galois group is the symmetric group $S_3$, the cyclic group $\mathbb {Z}/3\mathbb {Z}$, the cyclic group $\mathbb {Z}/2\mathbb {Z}$ or the identity when the degree is respectively 6, 3, 2, and 1.

• Step 7. If P(T) has a double root, then the type is $\hbox {I}^*_n$ with $n\ge 1$. Make a change of coordinates such that the double root is at the origin. Then $v(a_1)\ge 1$,    $v(a_2)= 1$, $v(a_3)\ge 2$,   $v(a_4)\ge 3$, ,   $v(a_6)\ge 4$, and $v(\Delta )=n+6$$(n\ge 1)$. We now assume that, except for their valuations, the Weierstrass coefficients are generic. We then distinguish between even and odd values of n.

  • (a) If $n=2\ell -3$ ($\ell \ge 2$), then $v(a_1)\ge 1$, $v(a_2)= 1$, $v(a_3)\ge \ell$,   $v(a_4)\ge \ell +1$,   $v(a_6)\ge 2\ell$, $v(b_6)=2\ell$, $v(b_8)=2\ell +1$, and

    $$\begin{aligned}T^2+a_{3,\ell } T -a_{6,2\ell } \end{aligned}$$

    has two distinct roots in its splitting field $\kappa '$. If the two roots are rational ($[\kappa ':\kappa ]=1$) then we have $\hbox {I}_{2\ell -3}^{*s}$ with dual graph $\widetilde{\text {D}}_{2\ell +1}$, otherwise ($[\kappa ':\kappa ]=2$) we have the fiber type $\hbox {I}_{2\ell -3}^{* ns}$ with dual graph $\widetilde{\text {B}}^t_{2\ell }$.

  • (b) If $n=2\ell -2$ ($\ell \ge 2$) then, $v(a_1)\ge 1$,    $v(a_2)= 1$, $v(a_3)\ge \ell +1$,   $v(a_4)\ge \ell +1$,   $v(a_6)\ge 2\ell +1$, and $v(b_8)=2\ell +2$. The polynomial

    $$\begin{aligned}a_{2,1} T^2+a_{4,\ell +1} T -a_{6,2\ell +1}\end{aligned}$$

    has two distinct roots in its splitting field. If the two roots are rational then we have $\hbox {I}_{2\ell -2}^{*s}$ with dual graph $\widetilde{\text {D}}_{2\ell +2}$, otherwise $\hbox {I}_{2\ell -2}^{* ns}$ with dual graph $\widetilde{\text {B}}^t_{2\ell +1}$.

• Step 8. If P(T) has a triple root, change coordinates such that the triple root is zero. Then $v(a_1)\ge 1$, $v(a_2)\ge 2$, $v(a_3)\ge 2$, $v(a_4)\ge 3$, $v(a_6)\ge 4$.

  Let

  $$\begin{aligned} Q(T)=T^2+a_{3,2} T -a_{6,4}. \end{aligned}$$

  If Q has two distinct roots ($v(b_6)=4$ or equivalently $v(\Delta )=8$) the type is $\hbox {IV}^*$.

  The split type depends on the rationality of the roots. If $b_{6,4}$ is a perfect square modulo s, the fiber is $\hbox {IV}^{*\text {s}}$ with dual graph $\widetilde{\text {E}}_6$, otherwise the fiber is $\hbox {IV}^{*\text {ns}}$ with dual graph $\widetilde{\text {F}}_4^t$. The split form can be enforced with $v(a_6)\ge 5$ and $v(a_3)=2$.

• Step 9. If Q has a double root, we change coordinates so that the double root is at the origin. Then:

  $v(a_1)\ge 1$,   $v(a_2)\ge 2$,   $v(a_3)\ge 3$,   $v(a_4)=3$,   $v(a_6)\ge 5\implies$ type $\hbox {III}^*$.

• Step 10. $v(a_1)\ge 1$,   $v(a_2)\ge 2$,   $v(a_3)\ge 3$,   $v(a_4)\ge 4$,   $v(a_6)= 5\implies$ type $\hbox {II}^*$.

• Step 11. Else $v(a_i)\ge i$ and the equation is not minimal. Divide all the $a_i$ by $s^i$ and start again with the new equation.