(Campbell and Zakharevich, 2022, p. 2)
(Campbell and Zakharevich, 2022, p. 3)
(Campbell and Zakharevich, 2022, p. 3)
(Campbell and Zakharevich, 2022, p. 3)
(Campbell and Zakharevich, 2022, p. 3)
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(Alexeev et al., 2019, p. 2) Definition of moduli of polarized [[K3 surfaces]] in terms of [[ample]] line bundles.
(Alexeev et al., 2019, p. 2) By [[Torelli]] it is a [quasiprojective](Unsorted/projective%20scheme.md) variety which is a global quotient. Discussion of the [[Baily-Borel]] compactification and [[toroidal compactifications]] in terms of an admissible [fan](Unsorted/toric%20varieties.md). See [[effective divisor]].
(Alexeev et al., 2019, p. 2) Discussion of the [[slc compactification]] in terms of [[stable pairs]] -- pairs with [[slc]] singularities and [[ample]] [[log canonical class]].
(Alexeev et al., 2019, p. 2) Motivating question: the boundary of BB and toroidal compactifications are easy to describe but not modular, while the slc compactification is modular but not easy to describe. Are there comparison maps?
(Alexeev et al., 2019, p. 2) Motivation from [[PPAVs]], see [[Voronoi fan]] and [[theta divisor]].
(Alexeev et al., 2019, p. 2) Main result, part 1. See [[K3 surface]], [[toroidal compactification]], [[admissible fan]].
(Alexeev et al., 2019, p. 3) Main result, part 2. See [[Coxeter fan]], [[semitoric compactification]], [[stable pair compactification]], [[Stein factorization]], [[normalization]], and [[Dynkin diagrams]].
(Alexeev et al., 2019, p. 3)
(Alexeev et al., 2019, p. 4)
Published: J. Algebraic Geom. 30 (2021), no. 2, 331\u2013405.
\nIn the paper under review the authors study pairs $\\left(Y, C+\\frac{1+\\varepsilon}{2} B\\right)$ which are log canonical but not Kawamata log terminal (klt) and such that $(Y, C)$ is log del Pezzo of index 2 ; that is, $-2\\left(K_{Y}+C\\right)$ is Cartier and ample. They classify such pairs and find several different families (shapes). For each, they describe the moduli stack and a compactification. Then they prove that for each shape the moduli space is proper and its normalisation is a quotient of a projective toric variety for a certain generalised Coxeter fan. Construction of moduli spaces which are toric varieties modulo a Weyl group existed for the root lattices $A_{n}, B_{n}$ and $C_{n}$. This paper provides constructions for $D_{n}$ and $E_{n}$.
\nReviewed by Enrica Floris
\n
(Alexeev and Thompson, 2019, p. 2) Defining the moduli space of (weighted) stable curves
(Alexeev and Thompson, 2019, p. 2) Context (some motivating results)
(Alexeev and Thompson, 2019, p. 2) Main result of this paper.
(Alexeev and Thompson, 2019, p. 2) Definition of surface pairs.
(Alexeev and Thompson, 2019, p. 2) Details of main results
(Alexeev and Thompson, 2019, p. 9)
(Alexeev and Thompson, 2019, p. 10)
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\n
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Includes bibliographical references.
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(Chambert-Loir et al., 2018, p. IX)
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(Chambert-Loir et al., 2018, p. 69) pushforward of the Lefschetz motive
(Chambert-Loir et al., 2018, p. 69) projection formula
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(Chambert-Loir et al., 2018, p. 72) Proof of this statement follows
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(Chambert-Loir et al., 2018, p. 74) [[spreading out]]
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(Chambert-Loir et al., 2018, p. 81) important examples of [[additive categories]]
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(Chambert-Loir et al., 2018, p. 84) $\\K$ for triangulated categories
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(Chambert-Loir et al., 2018, p. 85) [[Hodge structure]]
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(Chambert-Loir et al., 2018, p. 94) [[cyclotomic character]]
(Chambert-Loir et al., 2018, p. 125) $\\K_0(\\mcv\\slice k)\\invert{\\LL}$, the localization at the Lefschetz motive
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(Chambert-Loir et al., 2018, p. 469) Appendix on [[birational geometry]].
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(Chambert-Loir et al., 2018, p. 471) blowdown
(Chambert-Loir et al., 2018, p. 471) [[weak factorization]] theorem
(Chambert-Loir et al., 2018, p. 472) [[weak factorization]] theorem, relation to [[strong factorization]] conjecture
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(Chambert-Loir et al., 2018, p. 478) Appendix on [[formal geometry]]
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(Chambert-Loir et al., 2018, p. 483) algebra of [[convergent power series]]
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(Chambert-Loir et al., 2018, p. 489)
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(Chambert-Loir et al., 2018, p. 491)
(Chambert-Loir et al., 2018, p. 492)
(Chambert-Loir et al., 2018, p. 492)
(Chambert-Loir et al., 2018, p. 496)
(Chambert-Loir et al., 2018, p. 497)
Theorem to cite for the 3 conditions on the configuration of 10 points:
\n\u201c(10) The number of discrirninant conditions-infinite for the general point set P20-is finite for the PI0 of nodes of S, a set subject to three conditions and containing niine absolute constants. Any two discriminant coliditions whose signatures are congruent modulo 2 impose the same FOURTH condition on the ten nodes. The (\") conditions of type f0(i1i2) =0 the ('3) of type f1(ili2i3)= 0, the (1s) of type f2(i1i2i8i4i5i6)=O, the (17) of type f8(ii2i3i4i5i8i7i8) =0, and the (10) of type f4(M i2i3ii5,i6i7i8i9i10) =0, 496-2P'1(2P-1) (p=5) in all, exhaust the number of independent discriminant conditions. The members of this finite set of conditions are permuted under Crernona transformation like the odd theta characteristics under the group of ? 1 (\u201d (Coble, 1919, p. image 9)
\n
(Campbell et al., 2019, p. 1)
(Campbell et al., 2019, p. 1)
(Campbell et al., 2019, p. 2)
(Campbell et al., 2019, p. 2)
(Campbell et al., 2019, p. 2)
(Campbell et al., 2019, p. 3)
(Campbell et al., 2019, p. 4)
(Campbell et al., 2019, p. 4)
(Campbell et al., 2019, p. 4)
(Campbell et al., 2019, p. 5)
(Campbell et al., 2019, p. 5)
(Campbell et al., 2019, p. 6)
(Campbell et al., 2019, p. 6)
(Campbell et al., 2019, p. 7)
(Campbell et al., 2019, p. 7)
(Campbell et al., 2019, p. 7)
(Campbell et al., 2019, p. 8)
(Campbell et al., 2019, p. 8)
(Campbell et al., 2019, p. 8)
(Campbell et al., 2019, p. 8)
(Campbell et al., 2019, p. 9) Most important example of a Waldhausen category for this paper.
(Campbell et al., 2019, p. 9)
(Campbell et al., 2019, p. 9)
(Campbell et al., 2019, p. 10)
(Campbell et al., 2019, p. 10)
(Campbell et al., 2019, p. 10)
(Campbell et al., 2019, p. 10)
(Campbell et al., 2019, p. 11)
(Campbell et al., 2019, p. 11)
(Campbell et al., 2019, p. 12)
(Campbell et al., 2019, p. 14)
(Campbell et al., 2019, p. 16)
(Campbell et al., 2019, p. 17)
(Campbell et al., 2019, p. 17)
(Campbell et al., 2019, p. 17)
(Campbell et al., 2019, p. 17)
(Campbell et al., 2019, p. 18)
(Campbell et al., 2019, p. 18)
(Campbell et al., 2019, p. 18)
(Campbell et al., 2019, p. 18)
(Campbell et al., 2019, p. 19)
(Campbell et al., 2019, p. 19)
(Campbell et al., 2019, p. 24)
(Campbell et al., 2019, p. 25)
(Campbell et al., 2019, p. 29)
(Campbell et al., 2019, p. 29)
(Campbell et al., 2019, p. 30)
(Campbell et al., 2019, p. 30)
(Campbell et al., 2019, p. 31)
(Campbell et al., 2019, p. 31) Main result
(Campbell et al., 2019, p. 31)
(Campbell et al., 2019, p. 32)
(Campbell et al., 2019, p. 33)
(Campbell et al., 2019, p. 34)
(Campbell et al., 2019, p. 34)
(Campbell et al., 2019, p. 34)
(Campbell et al., 2019, p. 35)
(Campbell et al., 2019, p. 35)
(Campbell et al., 2019, p. 35)
(Campbell et al., 2019, p. 35)
(Campbell et al., 2019, p. 36)
(Fulton, p. 4)
(Hartshorne, 2008, p. 10)
(Hartshorne, 2008, p. 18)
(Hartshorne, 2008, p. 77)
(Hartshorne, 2008, p. 218)
(Hartshorne, 2008, p. 310)
(Hartshorne, 2008, p. 373)
\u201cwe treat two special classes of surfaces, the ruled surfaces, and the nonsingular cubic surfaces in P\u00b0,\u201d (Hartshorne, 2008, p. 373)
\n\u201cAs applications we give the Hodge index theorem and the Nakai-Moishezon criterion for an ample divisor.\u201d (Hartshorne, 2008, p. 373)
\n\u201cwe prove the theorem of factorization of a birational morphism i\u201d (Hartshorne, 2008, p. 373)
\n\u201cprove Castelnuovo9s criterion for contracting an exceptional curve of the first kind.\u201d (Hartshorne, 2008, p. 373)
\n\u201cHere the theory of curves gives a good handle on the ruled surfaces, because many properties of the surface arc closely related to the study of certain linear systems on the base curve.\u201d (Hartshorne, 2008, p. 373)
\n\u201cthere is a close connection between ruled surfaces over a curve C and locally free sheaves of rank 2 on C,\u201d (Hartshorne, 2008, p. 373)
\n\u201ce study the nonsingular cubic surfaces in P3, and the famous 27 1ineS which lie on those surfaces. By representing the surface as a P\u00b0 with 6 points blown up, the study of linear systems on the cubic surface is reduced\u201d (Hartshorne, 2008, p. 373)
\n\u201cto the study of certain linear system of plane curves with assigned base points.\u201d (Hartshorne, 2008, p. 374)
\n\u201che Riemann4 Roch theorem for surfaces gives a connection between the dimension of a complete linear system |D|, which is essentially a cohomological invariant, and certain intersection numbers on the surface.\u201d (Hartshorne, 2008, p. 374)
\n\u201csurface will mean a nonsingular projective surface over an algebraically closed field k.\u201d (Hartshorne, 2008, p. 374)
\n\u201ccurve on a surface will mean any effective divisor on the surface. In particular, it may be singular, reducible or even have multiple components. A point will mean a closed point, unless otherwise specified.\u201d (Hartshorne, 2008, p. 374)
\n\u201cIf C and D are curves on X, and if Pe C n D is a point of intersection of C and D, we say that C and D meet transversally at P if the local equations f,g of C,D at P generate the maximal ideal mp of Op x.\u201d (Hartshorne, 2008, p. 374)
\n\u201cPrOOF. We embed X in a projective space P\" using the very ample divisor D.\u201d (Hartshorne, 2008, p. 375)
\n\u201cThen we apply Bertini9s theorem\u201d (Hartshorne, 2008, p. 375)
\n
(Hartshorne, 2008, p. 375)
\u201cHere, of course, .#(D) is the invertible sheaf on X corresponding to D (I, \u00a77), and deg. denotes the degree of the invertible sheaf \u00a5 (D) \u00ae (\u00a2 on C (IV,\u00a71).\u201d (Hartshorne, 2008, p. 375)
\n\u201c#(4 D) is the ideal sheaf of D on X.\u201d (Hartshorne, 2008, p. 375)
\n\u201cf C and D are curves with no common irreducible component, and if P \u00ac C n D, then we define the intersection multiplicity (C.D)p of C and D at P to be the length of (. x/( f.\u00a2), where f,g are local equations of C.D at P (I, Ex 5.4). Here length is the same as the dimension of a k-vector space.\u201d (Hartshorne, 2008, p. 377)
\n
(Hartshorne, 2008, p. 377) Intersection number as a sum of local intersection multiplicities
\u201cIf D is any divisor on the surface X, we can define the selfintersection number D.D, usually denoted by D*. Evenif Cisa nonsingular curve on X, the self-intersection C? cannot be calculated by the direct metho\u201d (Hartshorne, 2008, p. 377)
\n
(Hartshorne, 2008, p. 378) How to compute self-intersection numbers as the degree of the normal sheaf $\\mcn_{C, X}$.
\u201ce must use linear equivalence.\u201d (Hartshorne, 2008, p. 378)
\n(Hartshorne, 2008, p. 378) Intersection numbers on P^2
\n(Hartshorne, 2008, p. 378) Intersection numbers of quadric surfaces in P^3
\n(Hartshorne, 2008, p. 378) Self-intersection of the canonical
\n
(Hartshorne, 2008, p. 378) Adjunction formula
(Hartshorne, 2008, p. 441)
\u201c1 Intersection Theory\u201d (Hartshorne, 2008, p. 442)
\n
(Hartshorne, 2008, p. 443)
\u201cIt is called the Chow ring of X.\u201d (Hartshorne, 2008, p. 443) The Chow ring.
\n\u201cProperties of the Chow Ring\u201d (Hartshorne, 2008, p. 445)
\n\u201c3 Chern Classes\u201d (Hartshorne, 2008, p. 446)
\n\u201cThe Riemann-Roch Theorem\u201d (Hartshorne, 2008, p. 448)
\n
(Hartshorne, 2008, p. 449) Hirzebruch-Riemann-Roch theorem.
(Hartshorne, 2008, p. 455)
(Hartshorne, 2008, p. 466)
(Hoekzema et al., 2022, p. 2)
(Hoekzema et al., 2022, p. 2)
(Hoekzema et al., 2022, p. 2)
(Hoekzema et al., 2022, p. 3)
(Hoekzema et al., 2022, p. 3)
(Hoekzema et al., 2022, p. 4)
(Hoekzema et al., 2022, p. 5)
(Hoekzema et al., 2022, p. 6)
(Hoekzema et al., 2022, p. 6)
(Hoekzema et al., 2022, p. 7)
(Hoekzema et al., 2022, p. 8)
(Hoekzema et al., 2022, p. 8)
(Hoekzema et al., 2022, p. 10)
(Hoekzema et al., 2022, p. 11)
(Hoekzema et al., 2022, p. 11)
(Hoekzema et al., 2022, p. 11)
(Hoekzema et al., 2022, p. 12)
(Hoekzema et al., 2022, p. 12)
(Hoekzema et al., 2022, p. 15)
(Hoekzema et al., 2022, p. 16)
(Hoekzema et al., 2022, p. 19)
(Hoekzema et al., 2022, p. 19)
(Hoekzema et al., 2022, p. 21)
Theory of Hermitian symmetric spaces
\nBuilding blocks for discrimiinant forms.
\nLattices for nodal Enriques surfaces
\nModuli space of nodal Enriques surfaces
\n
(Sarazola and Shapiro, 2021, p. 2)
(Sarazola and Shapiro, 2021, p. 2)
(Sarazola and Shapiro, 2021, p. 2)
(Sarazola and Shapiro, 2021, p. 3)
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(Sarazola and Shapiro, 2021, p. 26)
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(Sarazola and Shapiro, 2021, p. 37)
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(Sarazola and Shapiro, 2021, p. 37)
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(Sarazola and Shapiro, 2021, p. 38)
(Sarazola and Shapiro, 2021, p. 45)
(Sarazola and Shapiro, 2021, p. 45)
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(Sarazola and Shapiro, 2021, p. 47)
(Sarazola and Shapiro, 2021, p. 47)
(Sarazola and Shapiro, 2021, p. 47)
(Sarazola and Shapiro, 2021, p. 49)
(Zakharevich, 2016, p. 2)
(Zakharevich, 2016, p. 3)
(Zakharevich, 2016, p. 3)
(Zakharevich, 2016, p. 3)
(Zakharevich, 2016, p. 4)
(Zakharevich, 2016, p. 4)
(Zakharevich, 2016, p. 4)
(Zakharevich, 2016, p. 5)
(Zakharevich, 2016, p. 5)
(Zakharevich, 2016, p. 5)
(Zakharevich, 2016, p. 6)
(Zakharevich, 2016, p. 6)
(Zakharevich, 2016, p. 6)
(Zakharevich, 2016, p. 6)
(Zakharevich, 2016, p. 7)
(Zakharevich, 2016, p. 7)
(Zakharevich, 2016, p. 9)
(Zakharevich, 2016, p. 9)
(Zakharevich, 2016, p. 10)
(Zakharevich, 2016, p. 14)
(Zakharevich, 2016, p. 23)
(Zakharevich, 2016, p. 26)
(Zakharevich, 2016, p. 27)
(Zakharevich, 2017, p. 1)
\u201cThis ring is quite complicated; for example, it is not an integral domain\u201d (Zakharevich, 2017, p. 1)
\n
(Zakharevich, 2017, p. 1)
(Zakharevich, 2017, p. 1)
\u201cIn fact, Borisov\u2019s main result was to construct an element in the kernel of multiplication by L, and, seemingly coincidentally, his method also constructed an element in the kernel of \u03c8n.\u201d (Zakharevich, 2017, p. 1) Borisov's coincidence.
\n
(Zakharevich, 2017, p. 2)
(Zakharevich, 2017, p. 2) Theorem A
(Zakharevich, 2017, p. 2) Theorem B
(Zakharevich, 2017, p. 2) Theorem C
(Zakharevich, 2017, p. 2) Theorem D
(Zakharevich, 2017, p. 3) Theorem E. This morphism only takes *smooth* varieties to their single birational isomorphism class.
(Zakharevich, 2017, p. 3) Liu-Sebag's result
(Zakharevich, 2017, p. 3)
(Zakharevich, 2017, p. 3) Notation.
(Zakharevich, 2017, p. 4) Definition of assemblers.
(Zakharevich, 2017, p. 4) Fundamental theorem of $\\Asm$. Relations between generators left imprecise.
(Zakharevich, 2017, p. 4)
(Zakharevich, 2017, p. 4)
(Zakharevich, 2017, p. 5)
(Zakharevich, 2017, p. 5)
(Zakharevich, 2017, p. 5)
(Zakharevich, 2017, p. 5)
(Zakharevich, 2017, p. 5)
(Zakharevich, 2017, p. 5) #todo write as a tower with cofibers on the side.
(Zakharevich, 2017, p. 5) devissage for assemblers
(Zakharevich, 2017, p. 6)
\u201cTheorem A follows directly from several results in [ZakA]. Here, we give an outline of the proof by reducing of the theorem to those results.\u201d (Zakharevich, 2017, p. 6) Proof of theorem A
\n
(Zakharevich, 2017, p. 6)
(Zakharevich, 2017, p. 6)
(Zakharevich, 2017, p. 7) $\\Sigma \\cat{C}.$
(Zakharevich, 2017, p. 7) Cofiber of maps between assemblers
(Zakharevich, 2017, p. 7)
(Zakharevich, 2017, p. 7) Embedded exact sequences
(Zakharevich, 2017, p. 8)
(Zakharevich, 2017, p. 8)
(Zakharevich, 2017, p. 8)
(Zakharevich, 2017, p. 9)
\u201cProof.\u201d (Zakharevich, 2017, p. 9)
\n\u201cThus the boundary map in the long exact sequence associated to the inclusion of one filtration degree into the next measures the error of a birational automorphism of the variety extending to a piecewise automorphism\u201d (Zakharevich, 2017, p. 9)
\n
(Zakharevich, 2017, p. 9)
\u201cProof.\u201d (Zakharevich, 2017, p. 9)
\n
(Zakharevich, 2017, p. 10)
\u201cProof.\u201d (Zakharevich, 2017, p. 10)
\n
(Zakharevich, 2017, p. 10)
(Zakharevich, 2017, p. 11)
\u201cProof.\u201d (Zakharevich, 2017, p. 11)
\n
(Zakharevich, 2017, p. 14)
\u201cProof.\u201d (Zakharevich, 2017, p. 14)
\n\u201cProof of (2):\u201d (Zakharevich, 2017, p. 15)
\n\u201cProof of (3):\u201d (Zakharevich, 2017, p. 15)
\n
(Zakharevich, 2017, p. 15)
\u201cProof.\u201d (Zakharevich, 2017, p. 15)
\n
(Zakharevich, 2017, p. 15)
(Zakharevich, 2017, p. 16)
(Zakharevich, 2017, p. 16)
\u201cProof.\u201d (Zakharevich, 2017, p. 16)
\n
(Zakharevich, 2017, p. 16)
(Zakharevich, 2017, p. 16)
(Zakharevich, 2017, p. 17)
\u201cProof.\u201d (Zakharevich, 2017, p. 17)
\n
(Zakharevich, 2017, p. 18)
\u201cProof.\u201d (Zakharevich, 2017, p. 18)
\n
(Zakharevich, 2017, p. 18)
(Zakharevich, 2017, p. 19)
\u201cProof of Theorem E.\u201d (Zakharevich, 2017, p. 19)
\n
(Zakharevich, 2017, p. 19)
\u201cProof of Theorem C.\u201d (Zakharevich, 2017, p. 19)
\n\u201cProof of Theorem D.\u201d (Zakharevich, 2017, p. 20)
\n
(Zakharevich, 2017, p. 20) Conjecture
(Zakharevich, 2017, p. 21) Conjecture