--- sort: 800 title: Loojenga IAS flashcard: "Orals::Looijenga IAS" --- # Loojienga/IAS - [x] What is a cusp singularity? ✅ 2022-11-30 - [ ] Any singularity such the the exceptional divisor of the minimal resolution is a cycle of rational curves. ![](attachments/Pasted%20image%2020221129235442.png) - [x] What does it mean for a singularity to be **smoothable**, i.e. what is a **smoothing**? ✅ 2022-11-30 - [ ] For $X$ with isolated singularities, a flat family $\mcx\to \DD$ with $\mcx_0 \cong X$ and all nearby fibers $\mcx_t$ smooth. E.g. most plane curves are smoothable, not automatic for surfaces. Hypersurface singularities are always smoothable. There is no general criterion. There exists non-smoothable varieties: e.g. a cone over a 2-dimensional abelian variety which can't be a limit of smooth projective varieties. Alternatively, quotient singularities in dimension $\geq 3$, since these are rigid. Du val singularities are smoothable, since they are hypersurface type, replace $F(x,y,z)= 0$ with $F(x,y,z) = t$. - [x] What is a **rational surface**? ✅ 2022-11-30 - [ ] A 2-dimensional projective variety birationally equivalent to $\PP^2$; simplest in the Enriques-Kodaira classification. Always a repeated blowup of a minimal rational surface, one of $\PP^2$ or $\FF_n$ (the Hirzebruch surface). Suffices to show irregularity $q=0$ and plurigenus $p_2 = 0$. - [x] What is a **contraction**? ✅ 2022-11-30 - [ ] A surjective proper morphism $f:X\to Y$ between normal projective schemes where $f_* \OO_X \cong \OO_Y$, or equivalently the geometric fibers are connected. - [x] What is a **contractible curve**? ✅ 2022-11-30 - [ ] E.g. if starting with a proper normal algebraic surface $X$ and a curve $C$ in $X$, a contraction morphism $\pi: X\to Y$ where $\pi(C)$ is a point. - [x] What is an **anticanonical pair**? ✅ 2022-11-30 - [ ] A pair $(Y, D)$ where $Y$ is a rational surface and $D$ is a cycle of rational curves $D = \sum D_i$. - [x] What is a negative definite pair? ✅ 2022-11-30 - [ ] An ACP where $M_{ij} \da D_i \cdot D_j$ is negative definite, i.e. $\inp z{Mz} < 0$ for all $z \in \RR^n\smz$. - [x] What is a **toric pair**? ✅ 2022-11-30 - [ ] An ACP $(Y, D)$ where $Y$ is a toric variety and $D$ is its toric boundary. - [x] What is an exceptional curve of the first kind? ✅ 2022-11-30 - [ ] Rational curves $C$ isomorphic to $\PP^1$ with $C^2 = -1$. - [x] What is a **resolution**? ✅ 2022-11-29 - [ ] For $X$ singular, a proper morphism $\pi: W\to X$ where $W$ is smooth, $\pi\inv(X^\smooth) \subseteq W$ is dense and $\pi$ induces an isomorphism on the smooth locus. - [x] What is a **minimal resolution**? ✅ 2022-11-29 - [ ] Minimal $W$ is minimal iff whenever $W'\to V$ is another resolution, it factors as $W'\to W\to V$. - [ ] Exist uniquely in dimensions 1 and 2, but may not in higher dimensions. For a surface with an isolated singularity at 0: $\pi: W\to V$ is minimal iff $\pi\inv(0)$ does not contain any exceptional curves of the first kind. Any non-minimal resolution must contain a -1 curve. - [x] What is the hyperbolic Inoue surface? ✅ 2022-11-29 - [ ] From type $\rm{VII}_0$ in the classification of complex analytic surfaces, those $V$ with Betti number $b_1 = 1$ and Kodaira dimension $\kappa = -\infty$. Every curve $C_i$ in $X$ is a component of one of two cycles $D$ or $\hat D$ where $D+\hat D \in \abs{-K_X}$. Blow $D$ and $\hat D$ down to get $\bar V$ with two dual cusps $p, \hat{p}$, tracked by the germ $(\bar V, p, \hat{p}$). - [x] Discuss the **dual divisors** $D$ and $\hat{D}$. ✅ 2022-11-29 - [ ] For $(\bar V, p)$ the germ of a cusp, the exceptional divisor $D$ of a minimal resolution forms (by definition) a cycle of smooth rational transversely intersecting curves with $D_{i\mod n}\cdot D_{i+1 \mod n} = 1$, which is characterized by the diagonal of the intersection matrix $\vector d = \tv{d_1,\cdots, d_n}$. The hyperbolic Innoue surface has two cycles $D, \hat{D}$ which blow down to dual cusps $p, \hat{p}$. - [x] What is **Looijenga's conjecture**? ✅ 2022-11-29 - [ ] Known: If $\hat{D}$ contracts to a smoothable cusp singularity $\hat{p}$, then $D$ is the anticanonical divisor of some rational surface. Conjecture: the converse, i.e. if $D$ is the anticanonical divisor of some rational surface, then the cusp singularity associated to $\hat{D}$ is smoothable. - [x] Why is the converse to Looijenga's conjecture true? ✅ 2022-11-29 - [ ] Statement: If $\hat{D}$ contracts to a smoothable cusp singularity $\hat{p}$, then $D$ is the anticanonical divisor for some rational surface. Suppose $\hat{p}$ is smoothable, then Loojienga shows $\bar{V}$ has a universal deformation which is semi-universal for the germ $(\bar V, p, \hat{p})$, so one gets a deformation $\pi: \mcv \to \DD$ with $\mcv_0 = \bar V$ where $(\bar V, p)$ is constant and $(\bar V, \hat{p})$ is smoothed. The general fiber $\mcv_t$ is a surface with a cusp singularity $p_t$ (basically $p$); simultaneously smooth all $p_t$ to get $\mcy\to \DD$ where $\mcy_t$ is smooth and $\mcy_0$ is a partially contracted $V$ with a cusp $\hat{p}$. The general fibers $\mcy_t$ are the desired rational surfaces with anticanonical divisor $D$. - [x] What is a **pseudofan**? ✅ 2022-11-30 - [ ] For $(V, D)$ an ACP where $D = \sum_{i=1}^nD_i$ is a cycle, the pseudofanof $(V, D)$ is the triangulated IAS whose underlying triangulated surface is $\Cone(\Gamma(D))$, the cone over the dual complex of $D$. - [x] What are the two main **surgeries** on pseudofans? ✅ 2022-11-30 - [ ] Internal blowups and node smoothing. - [ ] Pics: ![](attachments/Pasted%20image%2020221130175123.png) - [x] What is an **almost toric fibration**? ✅ 2022-11-30 - [ ] A Lagrangian fibration $\mu: (X, \omega) \to S$ whose general fiber is a smooth $T^2$ which undergoes symplectic reduction at $\bd S$ where fibers in $S\interior$ degenerate to wedges of $S^2$ at finitely many specified points. The **almost toric base** generalizes the moment polytope and has an IA structure with $v\in S^\sing \iff \mu\inv(v)$ is singular. $\mu\inv(\bd S)$ is an anticanonical divisor on $X$. - [x] Disuss almost toric fibrations over a disc. ✅ 2022-11-30 - [ ] Represented by $\mu: (Y, D, \omega)\to S$ a symplectic anticanonical pair where components of $D$ map to $\bd S$. Doing internal blowups on aboundary component $P_i\subseteq \bd S$ corresponds to an internal blowup on $D_i$ in $Y$, similarly for node smoothing. - [x] What is a **stratified space**? ✅ 2022-11-30 - [ ] A topological space $B$ with a filtration $\emptyset = B_{-1} \subseteq B_0 \subseteq \cdots$ where each $B_k$ is closed and the sections $S_d(B) \da B_d\sm B_{d-1}$ are smooth $d\dash$dimensional manifolds and $B = \Union_{d\geq 0} B_d$. Say $B$ is dimension $n$ if $n$ is maximal such that $S_n(B) \neq \emptyset$, in which case $S_n(B)$ is the **top stratum**. - [x] What is a **Lagrangian torus fibration**? ✅ 2022-11-30 - [ ] A fibration $f: (X^{2n}, \omega) \to B$ which is a proper continuous map where $B$ is an $n\dash$dimensional stratified space and $f$ is a smooth submersion over the top stratum of $B$ with connected Lagrangian fibers, where the remaining fibers are connected stratified spaces with isotropic strata. - [x] What is an order $k$ **refinment** of an IAS? ✅ 2022-11-30 - [ ] For $S$ an IAS, the new IAS $S[k]$ gotten by post-composing the charts on $S$ with multiplication by $k$. - [x] What is a **corner blowup**? ✅ 2022-11-30 - [ ] For $(Y, D)$ an ACP, a blowup at a node of $D$. Yields anticanonical $\pi^* D - E$ - [x] What is an internal blowup? ✅ 2022-11-30 - [ ] For $(Y, D)$ an ACP, a blowup at $p\in D^\sing$. Yields anticanonical $\pi^* D - E$ - [x] How do IAS surgeries affect **charge**? ✅ 2022-11-30 - [ ] **Internal blowup** on $D_i: d_i\mapsto d_{i} + 1$, no other changes. $Q(D) \mapsto Q(D) + 1$. **$+1$ charge**. - [ ] **Corner blowup** on $D_i \intersect D_{i+1}$: $(\cdots, d_i, d_{i+1},\cdots)\mapsto (\cdots, d_i+1, 1, d_{i+1} +1,\cdots)$ and $Q(D) \mapsto Q(D)$ **$+0$ charge**. - [ ] **Node smoothing** on $D_{i} \intersect D_{i+1}$: $(\cdots, d_i, d_{i+1}, \cdots)\mapsto (\cdots, d_{i} + d_{i+1} - 2,\cdots)$. **$+1$ charge.** - [x] What is a **type III anticanonical pair** $(\mcx_0, D)$? ✅ 2022-11-30 - [ ] A surface $\mcx_0$ with a decomposition $\mcx_0 = \Union_{i=0}^n V_i$ such that... - [ ] **Central HIS and normality**: $V_0$ is the hyperbolic Inoue surface with cycles $D, \hat D$ and for $i > 0$ the normalizations $\tilde V_i\to V_i$ are smooth rational surfaces. - [ ] **ACP Double Curves**: For $D_{ij}$ an irreducible double curve of $\mcx_0$ lying on $V_{i} \intersect V_j$, let $D_i$ be the union of the double curves $D_{ij}$ contained in $V_i$ and let $\tilde D_i$ be its inverse image under normalization; then require $(\tilde V_i, \tilde D_i)$ to be an ACP. - [ ] **Triple point formula**: For $D_{ij}$ a double curve joining $V_i$ and $V_j$, then $$\qty{\ro{D_{ij}}{\tilde V_i}}^2 + \qty{\ro{D_{ij}}{\tilde V_j}}^2 = \begin{cases}-2 & D_{ij} \text{ smooth} \\0 & D_{ij} \text{ nodal} \end{cases}$$ - [ ] The dual complex $\Gamma(\mcx_0)$ is a triangulation of $S^2$. - [x] What is **$d\dash$semistability**? ✅ 2022-11-30 - [ ] $\Ext_{\OO_{\mcx_0}}( \Omega_{\mcx_0}^1, \OO_{\mcx_0}) \cong \OO_{\mcx_0^\sing}$ - [x] What is the **Friedman-Miranda criterion**? ✅ 2022-11-30 - [ ] The cusp singularity associated to $\hat D$ is smoothable iff there exists a type III ACP $(\mcx_0, D)$. - [x] What is a **nodal curve**? ✅ 2022-12-11 - [ ] A complete algebraic curve such that every point is either smooth or a node, where $p$ is a node iff it admits a neighborhood which is complex analytically isomorphic to $V(xy)$. - [x] What is the **dual complex** of a type III anticanonical pair? ✅ 2022-12-11 - [ ] For $(\mcx_0, D)$ an ACP, the triangulation of $S^2$ with vertices are the components of $V_i$, directed edges $e_{ij}$ for double curves $D_{ij}$, triangular faces are triiple points $T_{ijk}$. - [x] What is a **degeneration**? ✅ 2022-11-29 - [ ] A degeneration of $X$: a flat proper morphism $\pi: \mcx \to \DD$ of complex analytic spaces such that $\pi$ is smooth over $\DD\interior$, the general fibers $\mcx_t$ are deformation equivalent to $X$ for all $t\in \DD\interior$, and the monodromy action on $H^2(\mcx_t; \ZZ)$ is nontrivial and unipotent. - [x] What is a **maximally unipotent degeneration**? ✅ 2022-11-29 - [ ] Unipotent: $r-1$ is nilpotent; for matrices, $p(t) = (t-1)^n$ for some $n$, so all eigenvalues are 1. Index of unipotency: the minimal power $r$ such that $(M-\id)^{r-1}\neq 0$ but $(M-\id)^r=0$. Maximal: of maximal possible unipotency index; alternatively the limiting mixed Hodge structure is Hodge-Tate type (extensions of sums of $\ZZ(n)$). Concretely, the monodromy operator $T: H^n(\mcx_t; \QQ) \selfmap$ satisfies $(T-I)^{n+1} = 0$ but $(T-I)^n\neq 0$. - [x] What is a basis triangle? ✅ 2022-11-29 - [ ] A triangle $\Delta \subseteq \RR^2$ such that $\mathrm{Vert}(\Delta) \subset \ZZ^2$ and $\mu(\Delta) = 1/2$. - [x] What is a **triangulated integral affine structure with singularities**? ✅ 2022-11-29 - [ ] A real surface with boundary (possibly empty) $\Sigma_g$ equipped with a triangulation $T$ such that $\Sigma_g\sm \mathrm{Vert}(T)$ admits an atlas $\ts{(U_i, \phi_i: U_i\to \RR^2)}$ such that $\phi_i \circ \phi_j\inv \in \SL_2(\ZZ)\semidirect \ZZ^2$ and for each triangle $\Delta \in T$, the interior $\Delta\interior$ admits a chart to a *basis triangle*. - [x] What is a **nodal trade**? ✅ 2022-11-30 - [ ] Alternative terminology for node smoothing. - [x] What is the **SYZ conjecture**? ✅ 2022-11-29 - [ ] Gross, 2008: If $X, X\dual$ are mirror CYs, they admit a common cospanning fibration $X\to B \from X\dual$ whose fibers are special Lagrangians and the general fiber is a torus. There should also be a canonical map $X\dual_t\iso H^1(X_t; \RR/\ZZ)$ and $X_t\iso H^1(X\dual_t; \RR/\ZZ)$ (likely too strong to be true) - [x] What is the relation to **SYZ mirror symmetry**? ✅ 2022-11-30 - [ ] Generally about boundaries in a CY moduli space, particularly near especially "bad" degenerations: maximally unipotent degenerations (MUD) or large complex structure limits. One can find special Lagrangian tori on CYs near these limit points, and approaching the limit, expect to see a larger portion of the CY filled by tori. Toric degnerations are a special case of MUD. Expect the base $B$ of a special Lagrangian fibration to be the Gromov-Hausdorff limit of a sequence of CYs near MUD, and GH limit should be the dual intersection complex of the AG degeneration. The base is an affine manifold. If $\pi:\mcx \to \DD$ is such a degeneration of CYs and $\mcx$ is polarized (equipped with $\mcl\in \Pic(\mcx)$ relatively ample), should exists a dual polarization $\hat\mcl$ on $\hat\mcx$. - [x] What is **charge**? ✅ 2022-11-30 - [ ] $Q(Y, D):=12+\sum_{i=1}^n\left(d_i-3\right)=12+\sum_{i=1}^n\left(a_i-b_i\right)$ where $\mathbf{d}=\left(d_1, \ldots, d_n\right)=(a_1+3, \underbrace{2, \ldots, 2}_{b_1}, \ldots, a_k+3, \underbrace{2, \ldots, 2}_{b_k})$ and $a_i, b_i \geq 0$. All toric pairs have charge zero. - [x] How can one obtain $\hat{ \mathbf d}$ for $\hat D$ from $\mathbf d$? ✅ 2022-11-30 - [ ] Interchange the $a_i$ and $b_i$: $\hat{\mathbf{d}}=\left(\hat{d_1},\cdots, \hat{d_s} \right)=(b_1+3, \underbrace{2, \ldots, 2}_{a_1}, \ldots, b_k+3, \underbrace{2, \ldots, 2}_{a_k})$ - [x] Discuss $Q(D)$ and $Q(\hat D)$. ✅ 2022-11-30 - [ ] $Q(D) \da 12 + \sum (d_i - 3)$ and $Q(D) + Q(\hat D) = 24$. This follows from the formula $Q(D) = 12 + \sum(a_i - b_i)$ and that $\hat D$ swaps $a_i$ and $b_i$. - [x] Discuss **conservation of charge**. ✅ 2022-12-11 - [ ] For $(\mcx_0, D)$ a Type III ACP, $\sum Q(V_i, D_i) = 24$. - [x] What is a **nonsingular IAS**? ✅ 2022-11-30 - [ ] When the atlas of charts on an IAS for $S\sm\mathrm{Vert}(T)$ extends to $S$; if not, denote the points where this fails as $S^\sing$. - [x] What is the **developing map**? ✅ 2022-11-30 - [ ] For $S$ an IAS with singularities, take contractible opens $U \subset S\sm S^\sing$ with charts $\phi_U: U\to \RR^2$, uniquely defined up to an element of $\SL_2(\ZZ)\semidirect\ZZ^2$, and define $\Phi: \widetilde{S\sm S^\sing} \to \RR^2$ from the universal cover by gluing any charts $\phi_U, \phi_V$ which agree on $U\intersect V$. Equivalet to the data of a monodromy representation $\pi_1(S\sm S^\sing) \to \SL_2(\ZZ) \semidirect \ZZ^2$. - [x] How are ACPs related to toric pairs? ✅ 2022-11-30 - [ ] Prop 5.1: every ACP $(Y, D)$ is a sequence of node smoothings and internal blowups of a toric pair $(\bar Y, \bar D)$. - [x] What is the neative self-intersection of an edge in an IAS? ✅ 2022-11-30 - [ ] For $e_{ik}$ an edge in $\Delta \subset T$ for an IAS, letting $e_{ij}, e_{il}$ be the edges emanating from $v_i$ clockwise and counterclockwise to $e_{ik}$, define $d_{ik}$ to be the "negative self intersection" of $e_{ik}$ by $$d_{ik} e_{ik} = e_{ij} + e_{il}$$ The data of $d_{ik}$ for each edge $e_{ik}$ satisfying $d_{ik} + d_{ki} = 2$ for each $i, k$ uniquely determines a triangulated IAS. - [ ] What is a **toric model**? - [ ] Discuss the integral affine completion of a disc to a sphere. - [ ] What is the moment polygon? - [ ] What is a type III degeneration of K3 surfaces? - [ ] What is a double curve? - [ ] What is a node smoothing? - [ ] What are the local charts of an IAS? - [ ] What is the (negative) self intersection associated to an edge in an IAS? - [ ] Discuss the dual cusps arising from blowdowns. - [ ] What is a minimally elliptic surface singularity? - [ ] What is symplectic reduction? - [ ] What is the star of a vertex? - [ ] What are cusps called cusps? - [ ] What does it mean for a surface to be algebraic? - [ ] What is a semistable model? - [ ] What is $d_{ij}$? - [ ] What is an exceptional curve on an anticanonical pair $(Y, D)$? - [ ] Discuss smoothing of a node in $D$ for an anticanonical pair $(V, D)$. (Prop 2.5) - [ ] When does a surface $\mcx_0$ smooth to an anticanonical pair in a family $\mcx \to \DD$? - [ ] Being a Type III ACP $(\mcx_0, D)$ plus $d\dash$semistability. - [ ] What is a Legendre dual? - [ ] What is a symplectic anticanonical pair? - [ ] What is a rational curve? - [ ] What is the germ of a singularity? - [ ] What is Artin's contractibility theorem? - [ ] What is Kodaira dimension? - [ ] What is the linear system $|-K_V|$? - [ ] What is a universal deformation? - [ ] What is a semi-universal deformation? # Talk Outline - Motivation: singularities. - Examples: - ![](attachments/Pasted%20image%2020221130174004.png) ![](attachments/Pasted%20image%2020221130174100.png) - Blowups: - Discuss dual intersection graphs and cycles associated to minimal resolutions. - For $\pi: (\tilde X, E)\to (X, 0)$ a minimal (no -1 curves) resolution, define $\Gamma$ whose vertices are components of $E$, put $e_{ij}\da E_i \cdot E_j$ edges between $v_i$ and $v_j$, and attach the number $e_{ii} \da -E_i^2$ to each $v_i$. Note $\Gamma$ has no graph-theoretic cycles. Form a matrix $M_{ij} = e_{ij}$, by a theorem of Grauert-Mumford $M$ is negative definite. - Example: ? ![](attachments/Pasted%20image%2020221130173928.png) - Discuss cusp singularities and dual cusps - Discuss smoothability and Loojenga's conjecture. - Discuss proof outline. - Discuss the Friedman-Miranda criterion.