Exercise: Canonical divisor. Let X be a smooth projective variety. Name the divisor that records the zeros and poles of a rational volume form on X.
Solution. That divisor is the canonical divisor K_X. For each positive integer m the nonzero sections of O_X(mK_X) are the m-pluricanonical forms. Their images in projective space have a maximum dimension, the Kodaira dimension kappa(X). The uniform pluricanonical degree theorem states: for each integer d>=1 there is an integer m(d)>0 such that, over every algebraically closed field of characteristic zero, the complete system |m(d)K_X| defines the Iitaka fibration of every smooth integral projective d-fold X with kappa(X)>=0.
Iitaka 1971, On D-dimensions of algebraic varieties: ./Iitaka1971.pdf
    > The D-dimension of an algebraic variety is defined and studied.
Kodaira 1963, On compact complex analytic surfaces I: https://www.jstor.org/stable/pdf/2373157.pdf
    > Elliptic surfaces and their canonical bundles organize the fibre structure.

Exercise: Log discrepancies. For a normal projective variety X with an effective rational boundary B, what inequalities define log canonical and kawamata log terminal singularities in terms of discrepancies?
Solution. Log canonical means every divisorial discrepancy is at least -1; kawamata log terminal means every divisorial discrepancy is strictly greater than -1. The log canonical divisor is K_X+B. The companion index theorem states: for each dimension d and each coefficient set Phi satisfying the descending chain condition, there is one integer a(d,Phi)>0 such that every projective log canonical pair (X,B) of dimension d with coefficients in Phi and K_X+B rationally linearly trivial satisfies a(d,Phi)(K_X+B)~0, an integral principal divisor.
Birkar-Zhang 2016, Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs: ./s10240-016-0080-x.pdf
    > The effective Iitaka fibration conjecture is reduced to polarized pairs.
Hacon-McKernan-Xu 2014, ACC for log canonical thresholds: ./1208.4150v1.pdf
    > Log canonical thresholds satisfy the ascending chain condition.

Exercise: Birational general type. In the general-type case, what does it mean for a pluricanonical map to be birational?
Solution. It means the map separates two general points, so the variety is mapped birationally onto its image. Hacon-McKernan, Takayama, and Tsuji proved that a fixed multiple of K_X gives a birational map for every smooth projective variety of general type. Viehweg-Zhang proved effective Iitaka fibrations for bases of dimension two with constants governed by the general fibre. The present theorem generalizes this to every nonnegative Kodaira dimension; in lower Kodaira dimensions the general fibre of the Iitaka fibration contributes its own torsion, canonical index, and monodromy data.
Viehweg-Zhang 2009, Effective Iitaka fibrations: ./0707.4287v3.pdf
    > Effective Iitaka fibrations
Hacon-McKernan 2006, Boundedness of pluricanonical maps of varieties of general type: ./0504327v3.pdf
    > A bounded pluricanonical degree separates a bounded family.
Takayama 2006, Pluricanonical systems on algebraic varieties of general type: https://doi.org/10.1007/s00222-006-0503-2
    > Pluricanonical maps are birational in uniformly bounded degree.
Tsuji 2007, Pluricanonical systems of projective varieties of general type II: ./ojm44_03_12.pdf
    > The boundedness statement is completed for projective varieties of general type.

Exercise: Canonical bundle formula. For a fibration f:X->Z with K_X rationally pulled back from Z, what is the canonical bundle formula relation among K_X and divisors on Z?
Solution. It writes K_X as rationally equivalent to f^*(K_Z+B_Z+M_Z), where B_Z is the discriminant divisor recording singular fibres and M_Z is the moduli divisor recording the variation of the fibers. Ambro constructed the moduli b-divisor and the klt-trivial fibration theorem that supplies B_Z and M_Z.
Ambro 2005, The moduli b-divisor of an lc-trivial fibration: ./0308143.pdf
    > The moduli b-divisor of an lc-trivial fibration is constructed.
Fujino-Mori 2000, A canonical bundle formula: https://projecteuclid.org/journals/journal-of-differential-geometry/volume-56/issue-1/A-Canonical-Bundle-Formula/10.4310/jdg/1090347529.pdf
    > A canonical bundle formula relates the total space to the base.

Exercise: Three induction statements. What three index/degree assertions are interleaved in the induction?
Solution. They are I_n, the uniform pluricanonical-degree theorem in dimension n; L_n, the uniform normal log canonical index theorem in dimension n for every rational descending-chain coefficient set; and K_n, the zero-boundary klt case of L_n. The step assumes I_j and L_j for all j<n, proves K_n by contradiction, then obtains L_n by adjunction and a Stein-degree descent, and finally obtains I_n from the canonical bundle formula and polarized effective birationality.
Kollar 2013, Singularities of the Minimal Model Program: https://doi.org/10.1017/CBO9781139547895
    > Singularities of the Minimal Model Program
Kollar-Mori 1998, Birational Geometry of Algebraic Varieties: https://doi.org/10.1017/CBO9780511662560
    > Birational Geometry of Algebraic Varieties

Exercise: Denominator theorem. State the denominator theorem used to supply the moduli divisor Cartier denominator.
Solution. Fix n>=1 and assume L_j for every j<n. There are positive integers p_0 dividing p, depending only on n, such that if f:X->Z is a contraction with dim X=n, dim Z>0, X klt, and K_X rationally linearly equivalent to a pullback from Z, then there are a rational function psi on X and canonical bundle formula data satisfying K_X+(1/p_0)div(psi)=f^*D_Z, D_Z=K_Z+B_Z+M_Z, where B_Z has coefficients in a fixed rational descending-chain set, the generalized pair (Z,B_Z+M_Z) is generalized klt, and p times the moduli b-divisor is b-Cartier.
Birkar-Zhang 2016, Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs: ./s10240-016-0080-x.pdf
    > The effective Iitaka fibration conjecture is reduced to polarized pairs.

Exercise: Vertical divisor field. If f:X->Z is a contraction and a rational function on X has a vertical divisor, where does that function live?
Solution. It belongs to the base field. In the denominator proof the geometric generic fibre is klt of lower dimension with rationally trivial canonical divisor, so the lower-dimensional index theorem gives a common p_0 trivializing it. Cartierness and one-dimensionality of the trivializing section space descend by faithful flatness, so the same p_0 works over the function field of Z. Then psi is chosen so that p_0K_X+div(psi) has no horizontal component.
Hacon-McKernan-Xu 2014, ACC for log canonical thresholds: ./1208.4150v1.pdf
    > Log canonical thresholds satisfy the ascending chain condition.
Ambro 2005, The moduli b-divisor of an lc-trivial fibration: ./0308143.pdf
    > The generic direct image has rank one in the rank-one condition.

Exercise: Discriminant coefficient. Let t_P be the log canonical threshold of f^*P over the generic point of P. What is the discriminant coefficient of B_Z at P?
Solution. It is 1-t_P. Threshold ACC forces these coefficients into a fixed rational descending-chain set. For one prime divisor P on a smooth determination W, if alpha is the coefficient of the pullback of D_Z at P and t is the threshold computed with the crepant total-space data, then the denominator of M_W at P is the denominator of alpha+t.
Fujino-Mori 2000, A canonical bundle formula: https://projecteuclid.org/journals/journal-of-differential-geometry/volume-56/issue-1/A-Canonical-Bundle-Formula/10.4310/jdg/1090347529.pdf
    > A canonical bundle formula relates the total space to the base.

Exercise: Weight of forms. Define the weight w_z(omega) of a relative pluricanonical form omega of degree m at a curve parameter z.
Solution. w_z(omega)=inf_E [ord_E(omega wedge dz/z)+m]/[m ord_E(z)], where E runs over vertical divisorial valuations centered over c. On the generic fibre set phi=(theta/dz)^{p_0} psi_C. The crepant sliced equality gives w_z(phi)=alpha-1+t.
Fujino-Mori 2000, A canonical bundle formula: https://projecteuclid.org/journals/journal-of-differential-geometry/volume-56/issue-1/A-Canonical-Bundle-Formula/10.4310/jdg/1090347529.pdf
    > A canonical bundle formula relates the total space to the base.

Exercise: Weight rules. What three elementary rules govern the weights?
Solution. Finite extension of the fibre function field with the base fixed does not change the weight of a pulled-back form; a curve extension of ramification l multiplies the pulled-back weight by l; taking a tensor power leaves the weight unchanged, while multiplying a degree-m form by a base function a adds ord_z(a)/m. These rules convert degenerations on a semistable model into arithmetic statements about units and roots of unity.
Deligne 1970, Equations differentielles a points singuliers reguliers: https://link.springer.com/content/pdf/10.1007/BFb0061194.pdf
    > Equations differentielles a points singuliers reguliers

Exercise: Beauville-Bogomolov blocks. What is the singular Beauville-Bogomolov decomposition of the geometric generic klt fibre?
Solution. After a finite extension of the curve function field, a finite quasi-etale cover of that fibre is a product of an abelian block, if present, and irreducible Calabi-Yau or irreducible holomorphic symplectic blocks. Taking a Galois comparison cover R and enlarging the curve field preserves geometric integrity and separates the factor fields for a subgroup of index at most d!, where d is the dimension of R. On each factor R_i a volume generator eta_i is chosen with w_u(eta_i)=0 after equivariant semistable reduction.
Druel 2018, A decomposition theorem for singular spaces with trivial canonical class of dimension at most five: ./1606.09006v1.pdf
    > A decomposition theorem for singular spaces with trivial canonical class
Greb-Guenancia-Kebekus 2019, Klt varieties with trivial canonical class: holonomy, differential forms, and fundamental groups: ./1704.01408v5.pdf
    > Holonomy, differential forms, and fundamental groups
Horing-Peternell 2019, Algebraic integrability of foliations with numerically trivial canonical bundle: ./1710.06183v2.pdf
    > Algebraic integrability of foliations with numerically trivial canonical bundle

Exercise: Equivariant block splitting. How does a finite or semilinear action separate the Beauville-Bogomolov factor directions?
Solution. Pullback splitting T_R=oplus_iE_i has no homomorphisms between distinct summands because a slice is a connected quasi-etale cover of a nonabelian block with no vector fields or intermediate one-forms. The summand projections are central idempotents, permuted by the action, and a subgroup of index at most d! fixes them. Normalized Stein factors give quasi-etale covers R_i->B_i carrying regular factor automorphisms.
Druel 2018, A decomposition theorem for singular spaces with trivial canonical class of dimension at most five: ./1606.09006v1.pdf
    > A decomposition theorem for singular spaces with trivial canonical class
Greb-Guenancia-Kebekus 2019, Klt varieties with trivial canonical class: holonomy, differential forms, and fundamental groups: ./1704.01408v5.pdf
    > Holonomy, differential forms, and fundamental groups
Horing-Peternell 2019, Algebraic integrability of foliations with numerically trivial canonical bundle: ./1710.06183v2.pdf
    > Algebraic integrability of foliations with numerically trivial canonical bundle

Exercise: Root of unity action. Let g be a finite-order lift of base inertia to R and b=n!. If (g^b)^*eta_i=c_i eta_i and lambda_i=c_i(c), why is lambda_i a root of unity?
Solution. Equal weights force ord_u(c_i)=0, so each c_i is a unit and lambda_i is a root of unity. The invariance of the original form gives the inertia identity zeta^{b ord_u(a)} prod_i lambda_i^{p_0}=1. If one integer N kills every lambda_i, then l divides bN ord_u(a), so p_0 b N w_z(phi) is an integer, clearing the denominator of alpha+t.
Prokhorov 2021, Equivariant minimal model program: https://doi.org/10.1070/RM9990
    > Equivariant minimal model program
Kempf-Knudsen-Mumford-Saint-Donat 1973, Toroidal Embeddings I: https://link.springer.com/content/pdf/10.1007/BFb0070318.pdf
    > Toroidal Embeddings I

Exercise: Abelian block eigenvalues. For an abelian block of dimension s, why is the eigenvalue lambda of the normalized Hodge frame uniformly bounded?
Solution. The normalized form is a local frame of the extended Hodge line in degree s under semistable reduction. Its eigenvalue is an eigenvalue of an integral matrix obtained from monodromy and finite base rotation. The rank of the relevant integral local system is binomial(2s,s), so the cyclotomic degree of lambda is bounded by that rank. There are only finitely many possible orders for each s<n.
Steenbrink 1976, Limits of Hodge structures: https://link.springer.com/content/pdf/10.1007/BF01403146.pdf
    > Limits of Hodge structures
Deligne 1970, Equations differentielles a points singuliers reguliers: https://link.springer.com/content/pdf/10.1007/BFb0061194.pdf
    > Equations differentielles a points singuliers reguliers

Exercise: Nonabelian good model. For a nonabelian block, what good equivariant projective dlt model Y is constructed near the marked point?
Solution. One constructs a G-equivariant projective model Y such that Y is klt, the central fibre T=Y_c is reduced Cartier, (Y,T) is dlt, and K_Y is semiample over the curve. The existence of a preserved component follows from structure-sheaf cohomology and the coherent Lefschetz theorem. The relative log divisor of eta wedge du/u is effective and rationally trivial, forcing it to vanish; then eta wedge du/u is a degree-one log generator and its residue makes K_T trivial and provides K_S+Theta_S on every preserved component S of T.
Horing-Peternell 2019, Algebraic integrability of foliations with numerically trivial canonical bundle: ./1710.06183v2.pdf
    > Algebraic integrability of foliations with numerically trivial canonical bundle
Kollar 2023, Families of Varieties of General Type: ./modbook-final.pdf
    > Families of Varieties of General Type

Exercise: Base torsion. For a normal rationally connected base Z with D=K_Z+B+M_Z rationally linearly trivial, what integer makes ell D an integral principal divisor?
Solution. There is an integer ell(d,I,p) such that ell D is an integral principal divisor. The proof extracts marked valuations by a klt boundary perturbation, uses global ACC for a finite coefficient set, then applies Birkar's boundedness theorem. Kummer theory and SGA1 identify Cl(Z)[a] with Hom(H_1(U(C),Z),mu_a(C)), and a common multiple T of the finitely many exponents kills all torsion classes; since pM_Z is integral and qD is integral for a suitable q, the divisor TqD is principal.
Birkar 2025, Singularities on Fano fibrations and beyond: ./2305.18770v2.pdf
    > Singularities on Fano fibrations and beyond
SGA1 2003, Revetements etales et groupe fondamental: ./0206203.pdf
    > Revetements etales et groupe fondamental
Delfs-Knebusch 1982, On the homology of algebraic varieties over real closed fields: https://www.degruyter.com/document/doi/10.1515/crll.1982.335.122/pdf
    > On the homology of algebraic varieties over real closed fields

Exercise: High-index sequence. What rigid sequence reduces unboundedness of the zero-boundary klt index?
Solution. If the canonical indices were unbounded, there would be projective Q-factorial terminal n-folds V_i of canonical indices r_i tending to infinity such that each V_i has irregularity zero on a resolution and countable birational group; every positive-dimensional rational image of smaller dimension has a rationally connected resolution; every effective Weil divisor with positive Iitaka dimension is big; and the cyclic index cover Y_i->V_i is terminal with K_{Y_i}~0, while every equivariant rational fibration with positive-dimensional base and fibre has a general-type geometric generic fibre.
Hacon-McKernan-Xu 2014, ACC for log canonical thresholds: ./1208.4150v1.pdf
    > Log canonical thresholds satisfy the ascending chain condition.

Exercise: General-type fibre reduction. In the remaining high-index sequence, why must every intermediate equivariant fibration have a general-type fibre?
Solution. A general moving hyperplane on a descended rational image is big, and its pullback to the cover is big. On a smooth model its difference from the base-pullback is effective exceptional. Terminality makes K_U~_Q sum d_aE_a with d_a>0, so restricting to a smooth general fibre makes K_F big. Otherwise an effective Weil divisor of positive Iitaka dimension would not be big.
Kollar-Mori 1998, Birational Geometry of Algebraic Varieties: https://doi.org/10.1017/CBO9780511662560
    > Birational Geometry of Algebraic Varieties

Exercise: Single nonabelian block. Why does the reduction force an unbounded sequence to come from one nonabelian Calabi-Yau or symplectic block?
Solution. If the cover has at least two positive-dimensional blocks, the finite-character lemma bounds the canonical index. If the cover is purely abelian or has one big abelian block, the eigenvalue of the canonical character acts on H^n of an abelian variety, whose rank is binomial(2n,n), again giving a uniform bound. Thus only a single nonabelian Calabi-Yau or symplectic block remains.
Boucksom-Demailly-Paun-Peternell 2013, The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension: ./0405285v1.pdf
    > The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension
Graber-Harris-Starr 2003, Families of rationally connected varieties: ./0109220v1.pdf
    > Families of rationally connected varieties

Exercise: Gamma and epsilon. Define gamma(P;V) and epsilon(P;V) for an ample divisor P on V.
Solution. For a big nef semiample rational Cartier divisor P on V and a smooth point x, gamma(P;V) is the very general value of the normalized vanishing order of complete plurisections through x after placing V over a smooth projective model, and epsilon(P;V) is the very general Seshadri constant, the infimum of P*C/mult_xC over curves C through x. Asymptotic Riemann-Roch gives gamma(P;V)>=(P^n)^{1/n}.
Ein-Kuchle-Lazarsfeld 1995, Local positivity of ample line bundles: ./9408003v1.pdf
    > Local positivity of ample line bundles

Exercise: Chain and tracking. What do the generic-chain lemma and tracking lemma guarantee before the scalar estimates?
Solution. A finite invariant list of marked covering families has a rational fibration whose generic leaf has dimension h between 1 and dim V. If every generic member is a curve of cost at most B, a smooth resolution H of that leaf satisfies the complete-section bound P^hH<=(hB)^h. The tracking lemma creates a covering member of smaller positive dimension whenever d+1 spaced levels lie below gamma(N;Z).
Ein-Kuchle-Lazarsfeld 1995, Local positivity of ample line bundles: ./9408003v1.pdf
    > Local positivity of ample line bundles

Exercise: Scalar estimates. What is the scalar theorem for a normal projective canonical V with K_V rationally trivial and finite group G?
Solution. If P is an ample rational Cartier divisor with g^*P rationally equivalent to P for every g in G, and every G-equivariant intermediate rational fibration has a general-type fibre, then gamma(P;V)<=C_n(P^n)^{1/n} and epsilon(P;V)>=c_n(P^n)^{1/n}, with constants depending only on n.
Ein-Kuchle-Lazarsfeld 1995, Local positivity of ample line bundles: ./9408003v1.pdf
    > Local positivity of ample line bundles

Exercise: Small-volume lemma. Why does the small-volume lemma imply Kodaira-dimension-zero fibres for a sequence of smooth e-folds with L^e tending to 0?
Solution. The small-volume lemma says: if a sequence of smooth e-folds has L^e->0 and K_W L^{e-1}<=C'L^e, then after passing to Iitaka fibres their general fibres have fixed positive dimension, Kodaira dimension zero, and volume of K_U+L|_U tending to 0. The proof combines Fujino's weak positivity of direct images with birational flattening.
Fujino 2017, Notes on the weak positivity theorems: ./weak-posi11.pdf
    > Notes on the weak positivity theorems
Campana-Paun 2015, Orbifold generic semi-positivity: ./1303.3169v4.pdf
    > Orbifold generic semi-positivity
Mehta-Ramanathan 1982, Semistable sheaves on projective varieties and their restriction to curves: https://link.springer.com/content/pdf/10.1007/BF01450677.pdf
    > Semistable sheaves on projective varieties and their restriction to curves
Ramanan-Ramanathan 1984, Some remarks on the instability flag: https://projecteuclid.org/journals/tohoku-mathematical-journal/volume-36/issue-2/Some-remarks-on-the-instability-flag/10.2748/tmj/1178228852.pdf
    > Some remarks on the instability flag

Exercise: Diagonal jet bound. State the diagonal jet bound for diagonal quotients Z_t=Y^t/mu_{r,diag}.
Solution. For a normal projective canonical complex n-fold V with K_V rationally trivial and canonical index r, cyclic cover pi:Y->V, an ample rational Cartier divisor L with 1<=L^n<=2 and gamma(L;V)<=C_1, choose b_0 with 3(2b_0)^n<1/4 and b_0(C_1+1)<3/4. For every t>=2, P_t=theta_t^*L^{boxplus t} satisfies epsilon(P_t)<=(n+1)t^2/b_0. The proof compares two orders of vanishing of one determinant generated by ordinary jets specialized to large positive characteristic.
Langer 2004, Semistable sheaves in positive characteristic: https://doi.org/10.4007/annals.2004.159.251
    > Semistable sheaves in positive characteristic

Exercise: Truncated symmetric algebra. What is the truncated symmetric algebra A_bullet(E) used in the slope estimate?
Solution. A_bullet(E) is the symmetric algebra of a vector bundle E modulo the ideal generated locally by p-th powers of degree-one elements. On a rank-rho bundle, in a frame it is O[v_1,...,v_rho]/(v_1^p,...,v_rho^p). The truncated-power slope lemma gives mu_max(A_j(E))<=(p-1)deg E+rho(rho-1)G for all nonzero graded pieces.
Sun 2008, Direct images of bundles under Frobenius morphism: ./0611360v2.pdf
    > Direct images of bundles under Frobenius morphism
Kitadai-Sumihiro 2008, Canonical filtrations and stability of direct images by Frobenius morphisms: https://projecteuclid.org/journals/tohoku-mathematical-journal/volume-60/issue-2/Canonical-filtrations-and-stability-of-direct-images-by-Frobenius-morphisms/10.2748/tmj/1215442876.pdf
    > Canonical filtrations and stability of direct images by Frobenius morphisms
Mustata-Schwede 2014, A Frobenius variant of Seshadri constants: ./1203.1081v1.pdf
    > A Frobenius variant of Seshadri constants

Exercise: Two rank bounds. What two rank bounds are compared in the diagonal jet bound?
Solution. Ordinary jets at a very general point first produce a column map of full generic rank R on the diagonal product. After specialization to large characteristic, relative Frobenius turns the spanning sections into a column map of full generic rank R=p^{nt}. The diagonal rank is at most R/4 by slope estimates for the truncated symmetric algebra of the relative cotangent bundle. A nonzero maximal minor must therefore vanish to high order along the reduced diagonal.
Campana-Paun 2015, Orbifold generic semi-positivity: ./1303.3169v4.pdf
    > Orbifold generic semi-positivity
Mehta-Ramanathan 1982, Semistable sheaves on projective varieties and their restriction to curves: https://link.springer.com/content/pdf/10.1007/BF01450677.pdf
    > Semistable sheaves on projective varieties and their restriction to curves

Exercise: Movable curve contradiction. In the characteristic-zero movable curve test, what is D^+ and what contradiction arises?
Solution. On the blow-up of W at a very general x, define D^+=bL+(q+1)K_W and d_0=b(C_1+1). The maximal minor has a strictly smaller upper bound on order from the external-product lemma, but the diagonal-rank loss gives order at least 3R/4. The contradiction proves the diagonal jet bound with quadratic growth in t independent of r.
Boucksom-Demailly-Paun-Peternell 2013, The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension: ./0405285v1.pdf
    > The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension

Exercise: Degree-one forgetting. In the high-index contradiction, what polynomial bound forces a degree-one coordinate-forgetting map?
Solution. Low-cost curves on Z_t give a rational quotient of V^t with a geometric generic leaf H_t of dimension h_t>0 and degree bound P_t^{h_t}*H_t<=(h_tB)^{h_t}. For fixed T and sufficiently large r each single-slot projection is generically finite, so 1<=h_t<=n. Comparing lower and upper degree bounds gives d_t<=KT^{2n}, and choosing a constant stretch of length about T/n forces one adjacent projection to be birational. Permutation invariance gives a birational coordinate-forgetting map for every omitted coordinate.
Boucksom-Demailly-Paun-Peternell 2013, The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension: ./0405285v1.pdf
    > The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension

Exercise: Birational self-maps. How does the tangent distribution of the leaf produce birational self-maps of V?
Solution. The leaf tangent distribution projects isomorphically to each slot and defines a rational plane distribution A(x) and rational transport isomorphisms R(x,y) satisfying the cocycle relation. Choosing a reference point gives independent rational vector fields v_1,...,v_h. Their local holomorphic flows preserve the leaf quotient, so the rational inverse of the forgetting map expresses E_z(y) rationally in y. Thus each sufficiently small E_z is birational. Since the derivative of z->E_z(y_0) is nonzero, the image is uncountable, contradicting the countability of Bir(V).
Matsumura 1963, On algebraic groups of birational transformations: ./Matsumura1963.pdf
    > On algebraic groups of birational transformations

Exercise: Reconstructing L_n. How is the normal lc index theorem L_n reconstructed from K_n?
Solution. First, a crepant dlt modification plus a small boundary perturbation gives a klt pair whose canonical class is not pseudo-effective, so BCHM produces a Mori fibre space f:(X',B')->Z'. If Z' is a point, the variety is Fano type and Birkar's bounded complements theorem supplies a bounded complement and index. If the base has positive dimension, a component S of the horizontal boundary is used for divisorial adjunction; lower-dimensional L_{n-1}, the vertical-descent lemma, and adjunction give a common form degree q.
Birkar 2019, Anti-pluricanonical systems on Fano varieties: ./1603.05765v4.pdf
    > Anti-pluricanonical systems on Fano varieties
Birkar-Cascini-Hacon-McKernan 2010, Existence of minimal models for varieties of log general type: ./0610203.pdf
    > Existence of minimal models for varieties of log general type

Exercise: Stein degree bound. In the descent step, what degree bounds the field extension in Stein factorization?
Solution. Restricting a trivializing form to a normalized horizontal lc divisor S, taking residues, and using the vertical-descent lemma gives (deg h)D=div_Z(N_{k(Y)/k(Z)}a) after Stein factorization S^nu->Y->Z. The degree deg h is bounded by the arithmetic Stein-degree theorem applied to the generic log Calabi-Yau fibre, so a uniform multiple of K_X+B is principal.
Xu 2019, Some results about the index conjecture for log Calabi-Yau pairs: ./1905.00297v2.pdf
    > Some results about the index conjecture for log Calabi-Yau pairs
Hacon-McKernan-Xu 2014, ACC for log canonical thresholds: ./1208.4150v1.pdf
    > ACC for log canonical thresholds

Exercise: All-degree comparison. Why do the pluricanonical section spaces of a smooth X and its terminal good minimal model V agree in every degree?
Solution. On a common smooth resolution W, K_W=q^*K_V+A with A>=0 exceptional. An m-canonical divisorial form on V pulls back, after adding mA, to a regular form on W; conversely, a form regular on W is regular at every prime divisor of V. The same comparison between W and X identifies all degrees, so the Iitaka field descends exactly.
Kollar-Mori 1998, Birational Geometry of Algebraic Varieties: https://doi.org/10.1017/CBO9780511662560
    > Birational Geometry of Algebraic Varieties

Exercise: Final Iitaka step. How does the final uniform pluricanonical degree I_n follow from a good minimal model?
Solution. Let X be smooth projective of dimension n with kappa(X)>=0. A good minimal model V exists, and the semiample contraction of K_V represents the Iitaka fibration f:V->Z with K_V rationally equivalent to f^*D for a rational ample divisor D. If Z is a point, K_V~_Q0 and the index theorem gives a nonempty pluricanonical system. If dim Z>0, the denominator theorem provides p_0 and a big base adjoint rationally equivalent to D, so polarized effective birationality of Birkar-Zhang gives a uniform degree whose rounded base system is birational. The section comparison H^0(X,mK_X)=psi^{m/p_0}f^*H^0(Z,floor(mD_Z)) transfers the full Iitaka field to X. A common multiple over the finitely many possible base dimensions preserves all section ratios.
Birkar-Zhang 2016, Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs: ./s10240-016-0080-x.pdf
    > Polarized effective birationality supplies a uniform birational rounded system.

Exercise: All characteristic-zero fields. Why does the same integer work over every algebraically closed field of characteristic zero?
Solution. The variety, finitely many divisors, and a rational principalization of a sufficiently divisible multiple descend to a finitely generated subfield; extending scalars to C preserves smoothness, singularity conditions, section-space dimensions, and linear triviality. The uniform trivializing section on C descends because the relevant line bundle becomes trivial after extension, and field generation is checked after a sufficiently large divisible degree by the birational comparison over C. Hence the same integers m(d) and a(d,Phi) work over every algebraically closed characteristic-zero field.
Kollar-Mori 1998, Birational Geometry of Algebraic Varieties: https://doi.org/10.1017/CBO9780511662560
    > Birational Geometry of Algebraic Varieties
Donovan 1969, The Lefschetz-Riemann-Roch formula: ./BSMF_1969__97__257_0.pdf
    > The Lefschetz-Riemann-Roch formula
Greb-Kebekus-Kovacs-Peternell 2011, Differential forms on log canonical spaces: ./1003.2913v4.pdf
    > Differential forms on log canonical spaces

Exercise: Prime moduli denominator. How is the denominator of the moduli divisor bounded at one prime divisor P?
Solution. Resolve the total space over W with its crepant boundary and the inverse image of P. Intersect W with general very ample members until a smooth curve C remains, choosing a general point c in C cap P. Bertini applied to the finitely many resolution strata preserves the boundary coefficients and the SNC threshold calculation at c. The sliced equality is K_V+B^c+(1/p0)div(psi_C)=h^*D_C, with coeff_cD_C=alpha. By the Weight of forms exercise, w_z(phi)=alpha-1+t. By the Weight rules exercise and a finite function-field ramification index e, this is exactly the denominator of alpha+t, hence of M_W at P.
Fujino-Mori 2000, A canonical bundle formula: https://projecteuclid.org/journals/journal-of-differential-geometry/volume-56/issue-1/A-Canonical-Bundle-Formula/10.4310/jdg/1090347529.pdf
    > A canonical bundle formula relates the total space to the base.
Deligne 1970, Equations differentielles a points singuliers reguliers: https://link.springer.com/content/pdf/10.1007/BFb0061194.pdf
    > Equations differentielles a points singuliers reguliers

Exercise: Separating block fields. How are the block factor fields separated and normalized on a semistable model?
Solution. Take a Galois comparison cover R of the original fibre function field including all conjugates of the product cover, and replace the curve field by its relative algebraic closure in C(R). If l is the total ramification over z and g acts by zeta on the tangent line at the marked point, choose on each generic Stein block a volume generator eta_i. Its pullback to a smooth resolution has no horizontal poles, and on a semistable model the weight is an integer. Multiply eta_i by a power of u so that w_u(eta_i)=0. The product of the normalized forms has weight zero, since relative log top forms multiply without an additional power of u in the toroidal charts. The pullback of phi and the p0-th power of this product both generate the generic p0-canonical line on R, so their ratio is a base function a, and the weight rules give phi=a(wedge_i eta_i)^{p0} and l w_z(phi)=ord_u(a)/p0.
Druel 2018, A decomposition theorem for singular spaces with trivial canonical class of dimension at most five: ./1606.09006v1.pdf
    > A decomposition theorem for singular spaces with trivial canonical class
Greb-Guenancia-Kebekus 2019, Klt varieties with trivial canonical class: holonomy, differential forms, and fundamental groups: ./1704.01408v5.pdf
    > Holonomy, differential forms, and fundamental groups
Horing-Peternell 2019, Algebraic integrability of foliations with numerically trivial canonical bundle: ./1710.06183v2.pdf
    > Algebraic integrability of foliations with numerically trivial canonical bundle

Exercise: Clearing base torsion. How is the torsion over rationally connected bases cleared?
Solution. First extract marked valuations and small modifications to place the boundary coefficients in a finite set and obtain a uniform positive lower bound for generalized log discrepancies. Birkar's boundedness theorem then makes the regular loci of the bounded family have only finitely many possible finitely generated homology groups H_1(U(C),Z). For each Z, a rationally connected resolution has vanishing Picard variety, so Cl(Z) is finitely generated. Kummer theory and SGA1 give Cl(Z)[a]~H^1_et(U,mu_a)~Hom(H_1(U(C),Z),mu_a(C)); hence H_1(U(C),Z) is finite and a common multiple T of the finitely many exponents kills every torsion class. Since pM_Z is integral and qD is integral for a suitable q, the divisor TqD is principal.
Birkar 2025, Singularities on Fano fibrations and beyond: ./2305.18770v2.pdf
    > Singularities on Fano fibrations and beyond
SGA1 2003, Revetements etales et groupe fondamental: ./0206203.pdf
    > Revetements etales et groupe fondamental
Delfs-Knebusch 1982, On the homology of algebraic varieties over real closed fields: https://www.degruyter.com/document/doi/10.1515/crll.1982.335.122/pdf
    > On the homology of algebraic varieties over real closed fields

Exercise: Exact scalar inequalities. What are the exact formulas for the upper gamma and lower Seshadri estimates?
Solution. For a big nef semiample rational Cartier P, asymptotic Riemann-Roch gives gamma(P;V)>=(P^n)^{1/n}. If no upper constant C_n exists, rescale a sequence so that P^n->0 while gamma(P;V)>1. In dimension at least two choose d+1 equally spaced levels below the lower gamma bound; the parameter-differentiation chain lemma produces a covering family of smaller-dimensional members with controlled polarization volume and controlled canonical degree. The small-volume lemma says: if a sequence of smooth e-folds has L^e->0 and K_W L^{e-1}<=C'L^e, then after passing to Iitaka fibres their general fibres have fixed positive dimension, Kodaira dimension zero, and volume of K_U+L|_U tending to 0. For the lower bound, if epsilon(P) were arbitrarily small, a moving family of low-cost curves would create a chain leaf whose small degree conflicts with P^n>=1, and a section of large degree vanishing on a thickened fibre would have normalized order exceeding the gamma bound. Hence gamma(P;V)<=C_n(P^n)^{1/n} and epsilon(P;V)>=c_n(P^n)^{1/n}.
Ein-Kuchle-Lazarsfeld 1995, Local positivity of ample line bundles: ./9408003v1.pdf
    > Local positivity of ample line bundles
Fujino 2017, Notes on the weak positivity theorems: ./weak-posi11.pdf
    > Notes on the weak positivity theorems
Campana-Paun 2015, Orbifold generic semi-positivity: ./1303.3169v4.pdf
    > Orbifold generic semi-positivity
Mehta-Ramanathan 1982, Semistable sheaves on projective varieties and their restriction to curves: https://link.springer.com/content/pdf/10.1007/BF01450677.pdf
    > Semistable sheaves on projective varieties and their restriction to curves
Ramanan-Ramanathan 1984, Some remarks on the instability flag: https://projecteuclid.org/journals/tohoku-mathematical-journal/volume-36/issue-2/Some-remarks-on-the-instability-flag/10.2748/tmj/1178228852.pdf
    > Some remarks on the instability flag

Exercise: Frobenius rank formulas. What are the exact rank formulas in the Frobenius diagonal argument?
Solution. Set b=b0/t and q=2tr. Ordinary jets at a very general point give full rank onto the jet module; choose Q=k0bL Cartier. Products of N=floor(p/k0) of the fixed jet sections span the p-th power thick point. The column map has source slots a_i>=0 with sum a_i=q, and evaluates into E^{boxtimes t} with E=F_*O_W(B), B=NQ, and full rank R=p^{nt}. The diagonal-rank loss uses filtered pieces G_j=O_W(tB+pqK_W) tensor A_j((Omega_W^1)^{oplus(t-1)}). The determinant slot line bundles have normalized numerical class c_1(Q_i)/R=B/p+((p-1)/(2p)+lambda_i)K_W. The movable curve test uses D^+=bL+(q+1)K_W and d0=b(C1+1); it gives an order upper bound Rb0(C1+1)<3R/4, while the diagonal-rank loss gives order at least 3R/4, a contradiction.
Sun 2008, Direct images of bundles under Frobenius morphism: ./0611360v2.pdf
    > Direct images of bundles under Frobenius morphism
Kitadai-Sumihiro 2008, Canonical filtrations and stability of direct images by Frobenius morphisms: https://projecteuclid.org/journals/tohoku-mathematical-journal/volume-60/issue-2/Canonical-filtrations-and-stability-of-direct-images-by-Frobenius-morphisms/10.2748/tmj/1215442876.pdf
    > Canonical filtrations and stability of direct images by Frobenius morphisms
Mustata-Schwede 2014, A Frobenius variant of Seshadri constants: ./1203.1081v1.pdf
    > A Frobenius variant of Seshadri constants
Langer 2004, Semistable sheaves in positive characteristic: https://doi.org/10.4007/annals.2004.159.251
    > Semistable sheaves in positive characteristic
Boucksom-Demailly-Paun-Peternell 2013, The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension: ./0405285v1.pdf
    > The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension

Exercise: Uncountable birational maps. How does the chain leaf produce uncountably many birational transformations?
Solution. For fixed T and sufficiently large covering degree r, low-cost curves define leaves H_t with P_t^{h_t}H_t<=(h_tB)^{h_t}. Single-slot projections are generically finite, so 1<=h_t<=n and the projection degree satisfies d_t<=KT^{2n}. A long constant stretch yields one adjacent projection of degree one, so permutation invariance gives a birational coordinate-forgetting map. The tangent distribution projects isomorphically to each slot, giving a rational plane distribution A(x) and rational transport isomorphisms R(x,y) with cocycle R(y,z)R(x,y)=R(x,z). Choose independent rational vector fields v_1,...,v_h and their local flows E_z. Rational recovery gives E_z(y)=ev(E_z(u),xi(u,y)); since the derivative of z->E_z(y0) is nonzero, the resulting birational self-maps are uncountable, contradicting countability of Bir(V).
Matsumura 1963, On algebraic groups of birational transformations: ./Matsumura1963.pdf
    > On algebraic groups of birational transformations
Boucksom-Demailly-Paun-Peternell 2013, The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension: ./0405285v1.pdf
    > The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension