--- created: 2024-04-26T13:30 updated: 2024-04-26T13:30 --- # An Invitation to Modern Enumerative Geometry > [!WARNING] **Do not modify** this file > This file is automatically generated by scrybble and will be overwritten whenever this file in synchronized. > Treat it as a reference. ## Pages ### [[An Invitation to Modern Enumerative Geometry.pdf#page=7|An Invitation to Modern Enumerative Geometry, page 7]] > A natural bridge between classical and modern enumerative geometry is provided by the theory of torus localisation. > these lecture notes can be thought of as an introduction to equivariant cohomology and localisation. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=8|An Invitation to Modern Enumerative Geometry, page 8]] ### [[An Invitation to Modern Enumerative Geometry.pdf#page=16|An Invitation to Modern Enumerative Geometry, page 16]] > Witten’s Conjecture, > Said conjecture claimed that integrals of cotangent line classes overg,nare governed by the KdV (Korteweg-de-Vries) hierarchy, a pretty extraordinary link with the theory of integrable hierarchies that Witten made following the idea that matrix models (governed by the KdV hierarchy), should be related to quantum gravity. > One of the most important open problems in the subject is to understand whether Pixton’s set of relations contains all tautological relations. Confirming this would be the analogue, forg,n, of understanding Schubert Calculus. > Donaldson–Thomas theory, ### [[An Invitation to Modern Enumerative Geometry.pdf#page=17|An Invitation to Modern Enumerative Geometry, page 17]] > a generating function of curve counting invariants on a given Calabi–Yau 3-fold X can be seen, in naive terms, as the probability amplitude for the propagation of strings in X. > It is nevertheless reasonable to conjecture that all curve counting theories, at the end of the day, will turn out equivalent to each other > Donaldson–Thomas theory is not disconnected from Gromov–Witten theory. In fact, the ‘MNOP Conjecture’ by Maulik–Nakrasov–Okounkov–Pandharipande > essentially states that the two constellations are the same, > the main characters of Donaldson–Thomas theory, namely Hilbert schemes on 3-folds, and more generally moduli spaces of sheaves on 3-folds, > one of the classical questions of enumerative geometry was the enumeration of rational curves of degree d on the (general, say) quintic 3- fold X ⊂ P4, one of the most studied Calabi–Yau 3-folds. The case d = 1 was known classically, through Schubert Calculus: the answer is 2875 > The case d = 2, yielding the answer 609,250, was confirmed in 1986 by the work of Sheldon Katz > The answer in the case d = 3, where the answer is 317,206,375, appeared in 1995 in the work of Ellingsrud and Strømme ### [[An Invitation to Modern Enumerative Geometry.pdf#page=18|An Invitation to Modern Enumerative Geometry, page 18]] ### [[An Invitation to Modern Enumerative Geometry.pdf#page=20|An Invitation to Modern Enumerative Geometry, page 20]] > How do we know how many constraints we should put on our objects in order to expect a finite answer? In other words, how do we ask the right question? > construct a moduli1space  for the objects we are interested in, > compactify > impose dim conditions to expect a finite number of solutions, ### [[An Invitation to Modern Enumerative Geometry.pdf#page=21|An Invitation to Modern Enumerative Geometry, page 21]] > where deg: A∗→ A∗pt = Z is the pushforward on Chow groups along the structure morphism  → pt > which exists by compactness of > if a solution comes with multiplicity bigger than one, there usually is a good geometric reason for this, and we should not disregard it ### [[An Invitation to Modern Enumerative Geometry.pdf#page=22|An Invitation to Modern Enumerative Geometry, page 22]] > In this case, if the torus fixed locusT⊂  is compact, a sensible enumerative solution to > counting problem can be defined by means of the localisation formula, one of the most important tools in enumerative geometry > TxX = (mx/m2x) ### [[An Invitation to Modern Enumerative Geometry.pdf#page=23|An Invitation to Modern Enumerative Geometry, page 23]] > we certainly want our enumerative answer not to depend on small perturbations of the geometry of the problem. Why do we want that? Just because we are reasonable people: ### [[An Invitation to Modern Enumerative Geometry.pdf#page=24|An Invitation to Modern Enumerative Geometry, page 24]] > The conormal sheaf of a closed immersion of schemes X → M defined by an ideal ℐ ⊂Mis the quasicoherent4X-module > X/M= ℐ/ℐ > and the normal sheaf is itsX-linear dual, > X/M= ℋomX(ℐ/ℐ2,X). The sheavesX/MandX/Mare locally free (of rank d) when X → M is a regular immersion (of codimension d). Example 2.3.4 If X → M = Pris a hypersurface of degree d, then the ideal sheaf of X in Pris the invertible sheafPr(−d), soX/Pr=Pr(d)|X > Notation 2.3.5 Let X → M be a closed immersion. We set NX/M= SpecXSymX/M. It is naturally a scheme over X. > the Euler sequence 0 →Pr→Pr(1)⊕r+1→Pr→ 0, ### [[An Invitation to Modern Enumerative Geometry.pdf#page=26|An Invitation to Modern Enumerative Geometry, page 26]] > C.C can be seen as the degree of the normal bundleC/S=S(C)|Cto C in S. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=28|An Invitation to Modern Enumerative Geometry, page 28]] > One can see the virtual class as a way of ### [[An Invitation to Modern Enumerative Geometry.pdf#page=29|An Invitation to Modern Enumerative Geometry, page 29]] > has a well-defined virtual dimension dvirat any point p ∈ , > dvir= dim Tp− dim Ob > where Ob is part of the data (a perfect obstruction theory) defining [] > The The virtual fundamental class is a Chow (or homology) class > encoding the ‘deformation theory of points’ p ∈  in the sense that, heuristically, > picture, locally around p, as being cut out by dim Ob |pequations inside the Zariski tangent space Tp > There are just a handful of cases where integrating a cohomology class on the moduli space α ∈ A∗→ H2∗(, Q) against []viris accessible > The moduli space  is smooth. In ### [[An Invitation to Modern Enumerative Geometry.pdf#page=30|An Invitation to Modern Enumerative Geometry, page 30]] > The moduli space  has a torus action. > many (often badly behaved) moduli spaces turn out to have a virtual fundamental class. > the moduli space of stable maps > the moduli space MHX(α) of μ-stable torsion free sheaves with Chern character α on a smooth polarised 3-fold (X, H ) such that H0(X, K−1X) = 0 > the moduli space PHX(α) of stable pairs with Chern character α on a smooth polarised Calabi–Yau 3-fold (X, H ) > Gromov–Witten theory = intersection theory ong,n(X, β), Donaldson–Thomas theory = intersection theory on MHX(α), Pandharipande–Thomas theory = intersection theory on PHX(α). ### [[An Invitation to Modern Enumerative Geometry.pdf#page=31|An Invitation to Modern Enumerative Geometry, page 31]] > Let C be a smooth projective curve over C. A linear system on C is a pair (ℒ,V) where ℒ is a line bundle on C and V ⊂ H0(C, ℒ) is a linear subspace. If deg ℒ = d and dimCV = r + 1, then (ℒ,V) is said to be a grdon the curve C. Let v ∈ V \ 0 be a section, P ∈ C a point. One defines ordPv = dimCℒP/vP· ℒP∈ Z≥0 to be the order of vanishing of v at P, where ℒPis the stalk of ℒ at P. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=32|An Invitation to Modern Enumerative Geometry, page 32]] ### [[An Invitation to Modern Enumerative Geometry.pdf#page=33|An Invitation to Modern Enumerative Geometry, page 33]] ### [[An Invitation to Modern Enumerative Geometry.pdf#page=34|An Invitation to Modern Enumerative Geometry, page 34]] > The dimension dim X of a scheme X is the supre mum of the lengths  of the chains V0V1···  V⊂ X of irreducible closed subsets. If X is a scheme, then dim X = supidim Xi, where { Xi}iare the irreducible components of X (i.e. the maximal irreducible closed subsets of X). > If a topological space X is irreducible and Y ⊂ X is a closed subset such that dim X = dim Y , then X = Y > quasicompact if the preimage of every affine open subset of S is quasicompact. We say that f is locally of finite type if for every x ∈ X there exist Zariski open neighbourhoods x ∈ Spec A ⊂ X and f (x) ∈ Spec B ⊂ S such that f (Spec A) ⊂ SpecB and the induced ring homomorphism B → A is of finite type, i.e. A is isomorphic to a quotient of B[x1,...,xn] as a B-algebra, for some n. We say that f is of finite type if it is locally of finite type and quasicompact. > If X is locally noetherian and quasicompact, then it is called noetherian. An important property of noetherian schemes is that they have a finite number of irreducible components, or, more generally, of associated points > Show that a morphism from a noetherian scheme is quasicompact. A scheme of finite type over a field k, i.e. a k-scheme X → Spec k, is then noetherian. > for any noetherian ring R ### [[An Invitation to Modern Enumerative Geometry.pdf#page=35|An Invitation to Modern Enumerative Geometry, page 35]] > A morphism of schemes f : X → S is a closed immersion (resp. open immersion) if f induces a homeomorphism between X and a closed subset (resp. an open subset) of S, and the induced local homomorphism f#x:S,f (x)→X,xis surjective (resp. an isomorphism) for all x ∈ X. A closed (resp. open) subscheme of a scheme S is the image of a closed (resp. open) immersion. A morphism X → S is called an immersion if it can be factored as X → Y → S, where X → Y is an open immersion and Y → S is a closed immersion. All immersions are locally closed immersions, i.e. can be factored as a closed immersion followed by an open immersion.1 > a locally closed immersion X → S is an immersion as long as S is locally noetherian, in which case > A scheme X is reduced if for every point x ∈ X the local ringX,xis reduced, i.e. it has no nilpotent elements besides 0 ∈ > X,x, the additive identity. A scheme is integral if it is reduced and irreducible. > Let R = k[u, v]/(uv, v2). > X = Spec R. Show that the point x ∈ X corresponding to the maximal ideal (u, v) ⊂ R > unique point such thatX,xis not reduced. > nonreduced scheme > Dn= Spec k[t]/tn, > One can show that quasicompact reduced schemes are precisely those schemes for which the regular functions on them are determined by their values on points. The function 0 = t ∈ k[t]/tn ### [[An Invitation to Modern Enumerative Geometry.pdf#page=36|An Invitation to Modern Enumerative Geometry, page 36]] > i.e. the image of t under the canonical map k[t]  k[t]/tn, vanishes at the unique point of Dn, and yet it is not the zero function! > the universal property just described is dual to the universal property of the tensor product for algebras; > X = Spec A, Y = Spec B and S = Spec R, then X×SY = Spec(A⊗RB) B) canonically. > the fibre of f over s is the fibre product Xs= X ×SSpec k(s). > The map f is said to be separated (resp. qua siseparated) iffis a closed immersion (resp. quasicompact). > If S > locally noetherian and f : X → S is locally of finite type, then it is quasiseparated. A closed immersion is always of finite type, hence quasicompact, thus separated implies quasiseparated. > A morphism f : X → S is affine if the preimage of every affine open subscheme of S is affine. In fact, f : X → S is affine if and only if X is isomorphic (over S) to SpecS ### [[An Invitation to Modern Enumerative Geometry.pdf#page=37|An Invitation to Modern Enumerative Geometry, page 37]] > one recovers ? = f∗X, where f : X → S is the structure morphism. > An algebraic variety over k (or simply a k-variety) is a separated scheme of finite type over Spec k. > (namely integral, separated scheme of finite type over a field) > An affine variety is a k-scheme of the form Spec A, where A = k[x1,...,xn]/I for some ideal I > Note that A is reduced if and only if I is a radical ideal. Thus a separated k-scheme is an algebraic variety if it admits a finite covering by affine varieties. > affine morphisms are separated > An algebraic variety X is projective if it admits a closed immersion into projective space > A variety is quasiprojective if it admits a locally closed immersion in some projective space, i.e. it is closed in an open subset of some Pn > The rational normal curve of degree d is the image of the closed embedding ιd: P1→ Pddefined by (u : v) → (ud: ud−1v :···: uvd−1: vd). This is the d-th Veronese embedding, ### [[An Invitation to Modern Enumerative Geometry.pdf#page=38|An Invitation to Modern Enumerative Geometry, page 38]] > An algebraic k-variety X is finite ifX(X) is > finite dimensional k-vector space. In this case, we say that  = dimkX(X) is the length of X. > A fat point (of length ) over Spec k is a k-scheme of the form X = Spec A, where A is a local Artinian k-algebra with residue field k (such that dimkA = ). Alternatively, fat points can be seen as those k-schemes X such that the > Xred→ X → Spec k is the identity. > Indeed, the Zariski tangent space TxX of a scheme X at a point x ∈ X, which by definition is the k(x)-vector space (mx/m2x)∗ > be identified with Homx(D2, X) = { h: D2→ X | h(0) = x }, where 0 ∈ D2is the unique closed point of D2 > Xa= Spec k[x,y]/(y − x2, y − a), a ∈ k. For a = 0, this scheme consists of two reduced points, corresponding to the maximal ideals > For a = 0, we get X0= Spec k[x]/x2= D2 ### [[An Invitation to Modern Enumerative Geometry.pdf#page=39|An Invitation to Modern Enumerative Geometry, page 39]] > Give an example of a scheme X whose underlying topological space consists of finitely many points, and yet is not finite. > A stronger notion than separatedness is properness. A morphism f : X → S is proper if it is separated, of finite type, and universally closed. The latter means that for every base change T → S, the induced map X ×ST → T is topologically a closed map. The valuative criterion for proper morphisms says that a finite type morphism f is proper if and only if for every valuation domain A with fraction field K there exists exactly one way to fill in the dotted arrow in a commutative square > All closed immersions are proper. > A morphism is finite if it is both proper and affine. It is quasifinite if it is locally of finite type, quasicompact, and has finite fibres. > A proper quasifinite morphism of noetherian schemes is finite. Projective morphisms provide other examples of proper morphisms. > An important notion in moduli theory is flatness. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=40|An Invitation to Modern Enumerative Geometry, page 40]] > is flat if it is flat at every point x ∈ X. It is moreover called faithfully flat if it is flat and surjective. > 3.1.23 Consider the closed subscheme Z = Spec k[x,y]/(y − x2) ⊂ A2 > Consider the morphism Z → A1= Spec k[t] defined, at the level of rings, by t → y. Then f is flat (cf. Example 3.1.16). All flat morphisms are open maps, in particular they have open image. Faithfully flat morphisms are epimorphisms in the category of schemes. > u: A  A is a square zero extension if (ker u)2= 0. > Then f is unramified (resp.smooth, étale) if for any square zero extension A  A of fat points over k and for any solid diagram > there exists at most one (resp. at least one, exactly one) way to fill in the dotted > étale = smooth + unramified. > A quasicompact integral scheme X is normal if for every closed point x ∈ X the local ringX,xis normal (i.e. integrally closed in its field of fractions). > A normalisation of X is a pair (Y, μ), where Y is a normal scheme and μ: Y → X is a morphism such that if μ: Y→ X is a dominant morphism (i.e. μhas dense image) from a normal scheme Y, then there exists a unique morphism θ : Y→ Y such that μ ◦ θ = μ ### [[An Invitation to Modern Enumerative Geometry.pdf#page=41|An Invitation to Modern Enumerative Geometry, page 41]] > A finite morphism f : X → S to a locally noetherian scheme is flat if and only if f is affine and f∗Xis finite locally free. > A morphism X → S of finite type k-schemes is smooth (of relative dimension d) if and only if it is flat and the fibres Xs→ Spec k(s) are smooth (of pure dimension d) for every s ∈ S. See > A morphism X → S of nonsingular k-varieties is smooth if and only if the tangent maps > X,x→ > Y,f (x)are surjective for all x ∈ X. > A morphism X → S of finite type k-schemes, where X is Cohen–Macaulay > S is smooth, is flat whenever the fibres have the same dimension. This is miracle flatness—see > A proper morphism X → Y of varieties which is injective on points and on tangent spaces is a closed immersion. In fact, a morphism of finite type between locally noetherian schemes is a closed immersion if and only if it is a proper monomorphism. > Let X be an integral scheme, S a normal locally noetherian scheme, f : X → S a proper birational morphism. ThenS→ f∗Xis an isomorphism, and there is > subset V ⊂ S (whose complement has codimension at least 2) such that f−1(V ) → V is an isomorphism and Xs= f−1(s) has no isolated points for all s ∈ S \ V > Let S be a normal, locally noetherian integral scheme, and let f : X → S be a separated, quasifinite, birational morphism of finite type. Then f is an open immersion. This is one of the many formulations of Zariski’s Main Theorem > A morphism of smooth C-varieties inducing an isomorphism on tangent spaces is étale. > An étale injective (resp. bijective) morphism is an open immersion (resp. an isomorphism). The following are all consequences of Zariski’s Main Theorem: > A birational proper morphism X → S of noetherian integral schemes has connected fibres whenever S is normal. > Let X, S be noetherian integral schemes, with S normal. A bijective, birational, proper morphism X → S is an isomorphism. > A finite (or even integral) birational morphism f : X → S of integral schemes with S normal is an isomorphism ### [[An Invitation to Modern Enumerative Geometry.pdf#page=42|An Invitation to Modern Enumerative Geometry, page 42]] > there are a bunch of points that are more relevant than all other points, in the sense that they reveal part of the behaviour of the structure sheaf: these points are the associated associated points of X. > A prime ideal p ⊂ R is said to be associated to M if p = AnnR(m) for some m ∈ M. The set of all associated primes is denoted3 APR(M) = { p | p is associated to M }. > Let p ⊂ R be a prime ideal. Then p ∈ APR(M) if and only if R/p is an R-submodule of M. > The minimal elements (with respect to inclu sion) in the set { p ⊂ R | p ⊃ AnnR(M) } are called isolated primes of M. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=43|An Invitation to Modern Enumerative Geometry, page 43]] > The non-isolated primes in APR(M) are called the embedded primes of M. > that Mi/Mi−1= R/pifor some prime ideal p > These primes are precisely the elements of APR(M). > Any ideal I ⊂ R has a primary decomposition, > I = q1∩···∩ q > of primary ideals. A proper ideal q  R is called primary if whenever a product xy lies in q, either x or a power of y lies in q. Put differently, every zero-divisor in R/q is nilpotent. > the radical of a primary ideal is prime, > q is p-primary if√q = p. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=44|An Invitation to Modern Enumerative Geometry, page 44]] > For an ideal I ⊂ R, one often calls the associated primes of I the associated primes of R/I . The minimal primes above I = AnnR(R/I ) (i.e. containing I ) correspond to the irreducible components of the closed > Spec R/I ⊂ Spec R, whereas for every embedded prime p ⊂ R there exists a minimal prime psuch that p⊂ p. Thus p determines an embedded component—a subvariety V (p) embedded in an irreducible component V (p). If the embedded prime p is maximal, we talk about an embedded point (Fig. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=45|An Invitation to Modern Enumerative Geometry, page 45]] > All sheaves on a scheme X in this text will be sheaves ofX-modules, > F ∈ ModXis free if it is isomorphic to⊕I > A locally free sheaf of rank r on X is a sheaf F such that there exists an open covering X =iUifor which F|Uiis free of rank r, for all i. > A quasicoherent sheaf on a scheme X is an > X-module F such that every point x ∈ X has an open neighbourhood U ⊂ X on which there is > ⊕IXU→⊕JXU→ FU→ 0 > A quasicoherent sheaf F is coherent if > is finitely generated, i.e. every point x ∈ X has an open neighbourhood U ⊂ X such that there is a surjective morphism⊕nX|UF|Ufor some positive integer > and > for any open subset U ⊂ X, for any positive integer n, and for any morphism s:⊕nX|U→ F|U, the kernel of s is finitely generated. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=46|An Invitation to Modern Enumerative Geometry, page 46]] > Let f : X → S be > The diagonalf: X → X ×SX is a locally closed immersion, > can find an open subscheme U ⊂ X ×SX and a closed immersion i : X → U. > ⊂Ube the corresponding ideal sheaf > Thenf= i∗(?/2) = i∗X/U (also denoted > X/S) is the (quasicoherent) sheaf of relative differentials (or relative cotangent sheaf) associated to f > It is coherent whenever f is of finite type and S is noetherian > If X = Spec k, where k is a field, then Coh X is equivalent to the category of finite dimensional k-vector spaces. > F∗= ℋomX(F,X) > F∨= RℋomX(F,X) for the derived dual of > F∗= h0(F∨), > ℰxtiX(F, E) = hi(RℋomX(F, E)), > Recall the correspondence between locally free sheaves on X and algebraic vector bundles V → X, sending F to the X-scheme π : V(F ) = SpecXSym F∗→ X. > the global sections H0(X, V(F )) of V(F ) > correspond toX-linear homomorphismsX→ F, > x ∈ X is naturally identified to the sheaf-theoretic fibre F (x) = Fx/mx· Fx= Fx⊗X,xk(x). ### [[An Invitation to Modern Enumerative Geometry.pdf#page=47|An Invitation to Modern Enumerative Geometry, page 47]] > T∗X = V(X), T X = V(X) the total spaces of the locally free sheavesXandX= ℋomX(X,X). > a 1-form on X is an element of H0(X, T∗X) = HomX(X,X). > Proj Sym(−) = P(−), > f : X → S is said to be projective if there exists a quasicoherent sheaf F on S such that f factors > X Proj Sym F S > X → Y → S, where X → Y is an open immersion and Y → S is projective. > An invertible sheaf ℒ on a scheme X is called ample if X is quasicompact and every point y ∈ X has an affine open neighbourhood of the form Xs= { x ∈ X | sx= 0 }, where s ∈ H0(X, ℒ⊗n) for some n > 0. > An invertible sheaf ℒ on X is f -ample if f is quasicompact and for every open affine subset U ⊂ S the line bundle ℒ|f−1 > is ample > Let f : X → S be a morphism. An invertible sheaf ℒ on X is f -very ample is there is a quasicoherentS-module F and a locally closed immersion ι: X → P(F ) over S such that ℒ∼=ι∗P(F )(1), ### [[An Invitation to Modern Enumerative Geometry.pdf#page=48|An Invitation to Modern Enumerative Geometry, page 48]] > coherent sheaf F ∈ Coh X is torsion free if for every x ∈ X the stalk F > is torsion free as anX,x-module, i.e. if for any a ∈X,x\ 0 the map Fx→ F > defined by τ → aτ is injective. Note that F is torsion free if and only if for every affine open subscheme U ⊂ X, theX(U )-module F (U ) is torsion free. > Its kernel is the torsion subsheaf of F, i.e. the subsheaf T (F ) ⊂ F whose sections over U ⊂ X are given by those elements τ ∈ F (U ) such that there exists a nonzero a ∈X(U ) for which aτ = 0 ∈ F (U ). > F is torsion free if and only if T (F ) = 0, if and only if νFis injective. > Then F ∈ Coh X is called reflexive if νFis an isomorphism. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=49|An Invitation to Modern Enumerative Geometry, page 49]] > On a regular scheme X, a reflexive sheaf is locally free away from a closed subset of codimension at least 3. > A reflexive sheaf of rank 1 on a smooth irreducible k-variety is a line bundle. > Let F be a coherent sheaf on X. The determinant of F is the line bundle detF =∧rk FF > ∗∗ > Let X be an arbitrary scheme. If ι: Z → X is a closed subscheme, then ℐZ= ker(Xι∗Z), called the ideal sheaf attached to ι, is quasicoherent (and coherent if X is locally noetherian). > In codimension 1, there is a bijection between the set of effective Cartier divisors on a scheme X and the isomorphism classes of pairs (ℒ, s), where ℒ is an invertibleX-module and s ∈ H0(X, ℒ) is a regular section, i.e. the associated mapX→ ℒ is injective. > D → (X(D), sD), whereX(D) = ℐ∗Dand sDis the image of 1 ∈Xunder the canonical mapX→X(D). > a pair (ℒ, s) defines the effective Cartier divisor D = Z(s) ⊂ X. > Given a closed immersion ι: Z → X as above, the functor ι∗: QCoh Z → QCoh X induces an equivalence between CohZ and the subcat egory of Coh X of coherentX-modules annihilated by ℐZ > If f : X → Y is a quasicompact morphism of schemes, the scheme-theoretic image of f is the smallest closed subscheme im(f ) of Y through which f factors. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=50|An Invitation to Modern Enumerative Geometry, page 50]] ### [[An Invitation to Modern Enumerative Geometry.pdf#page=244|An Invitation to Modern Enumerative Geometry, page 244]] > Let Artkdenote the category of local Artin k-algebras (A, mA) with residue field A/mA= k. Its opposite category is equivalent, via the Spec functor, to the category of fat points > i.e. the category of k-schemes S such that the structure morphism Sred→ Spec k is the identity. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=245|An Invitation to Modern Enumerative Geometry, page 245]] > ι: X → Y a closed k-subvariety. > E be a coherent sheaf > ϑ : E  Q be a surjection in Coh Y > In Definition A.1.1, one should think of  as the geometric object, defined over Spec k, that one wants to deform. Similarly, elements of D(A) should be interpreted > we took  to be the closed immersion ι: X → Y (resp. the sheaf E, the quotient ϑ : E  Q). > given a surjection φ : B  A in Artk, > Which elements of D(A) lift to D(B)? In other words, what is the image of φ∗: D(B) → D(A)? ### [[An Invitation to Modern Enumerative Geometry.pdf#page=246|An Invitation to Modern Enumerative Geometry, page 246]] > In fact, one can restrict attention to those surjections called small extensions. > A surjection B  A in Artkwith kernel I is called > a square zero extension if I2= 0, > a semi-small extension if I · mB= 0, > a small extension if it is semi-small and dimkI = 1. > every surjection B  A in Artkfactors as a composition of finitely many small extensions. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=247|An Invitation to Modern Enumerative Geometry, page 247]] > DerR(P , N) = HomP(P /R,N) for every P-module N. > Let ? be a nonempty set, G a group and σ : G×? → > an action of G on ?. The set ? is said to be a torsor (or principal homogeneous space) under the action of G if such action is transitive (i.e. there is just one orbit) and free (i.e. σ (g, s) = s implies g = 1 for all s ∈ S). In other words, ? is a torsor under G if the map G × ? → ? × ?, (g, s) → (σ (g, s), s) is bijective. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=248|An Invitation to Modern Enumerative Geometry, page 248]] > there is a natural obstruction to the existence of k-morphisms T → X making the diagram commute. Such obstruction lives in the cokernel of the natural map of k-vector spaces Homk(ι∗xi∗π,I) → Homk(ι∗xJ/J2, I ). ### [[An Invitation to Modern Enumerative Geometry.pdf#page=250|An Invitation to Modern Enumerative Geometry, page 250]] > A tangent-obstruction theory on a deformation functor D: Artk→ Sets is defined to be a pair (T1, T2) of finite dimensional k-vector spaces such that for any small extension I → B  A in Artkthere is an ‘exact sequence of sets’ T1⊗kI←→ D(B)←→ D(A) T←→2⊗kI, > Exactness at D(A) means that an element α ∈ D(A) lifts to D(B) if and only if ob(α) = 0. Exactness at D(B) means that, if there is a lift, then T1⊗kI acts transitively on the set of lifts. > The tangent space of the tangent-obstruction theory is T1, and is canonically determined by the deformation functor as T1= D(k[t]/t2). > (Prorepresentable) A deformation functor is prorepresentable if it is isomorphic to hR= Homk(R, −) for some local k-algebra R ∈ Lock. > If D: Artk→ Sets is prorepresentable by R, > setting T1(R) = TR= > If X → Y is a local complete intersection subscheme inside a projective k-variety Y , then Ti= Hi−1(X,X/Y), i = 1, 2, > If X → Y is an arbitrary closed subscheme, HX/Yhas the tangent-obstruction theory Ti= Exti−1Y(ℐX,X), i = 1, 2. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=251|An Invitation to Modern Enumerative Geometry, page 251]] > Ti= ExtiY(E, E), i = 1, 2, ### [[An Invitation to Modern Enumerative Geometry.pdf#page=252|An Invitation to Modern Enumerative Geometry, page 252]] > H0(X,X/Y)→ Ext1Y(ℐX, ℐX). ### [[An Invitation to Modern Enumerative Geometry.pdf#page=253|An Invitation to Modern Enumerative Geometry, page 253]] > H1(X,X/Y) → Ext2Y(ℐX, ℐX)/F2Ext2(ℐX, ℐX) = Ext2Y(ℐX, ℐX), > We now focus on examples related to real life: moduli spaces. Let M: Schopk→ Sets be a functor represented by a k-scheme M. Fix a point p ∈ M(k) = M(Spec k). > Defp(A) = { η ∈ M(Spec A) | ηk= p } where where ηkdenotes the image of η along the restriction map M(Spec A) → M(Spec k) induced by Spec k = Spec A/mA⊂ Spec A. > Defpis prorepresentable by the complete k-algebra R =;M,p, ### [[An Invitation to Modern Enumerative Geometry.pdf#page=254|An Invitation to Modern Enumerative Geometry, page 254]] > showing that if T2= 0 then M is smooth at p. The converse is not true > Let C ⊂ P2be a curve of degree d. Then ℐC= ?(−d) soC/P2= ### [[An Invitation to Modern Enumerative Geometry.pdf#page=256|An Invitation to Modern Enumerative Geometry, page 256]] > The fundamental class of X is the (possibly inhomogeneous) cycle [X] ∈ Z∗X determined by the irreducible components Vi⊂ X and their geometric multiplicities mi= lengthX,ξiX,ξ ### [[An Invitation to Modern Enumerative Geometry.pdf#page=257|An Invitation to Modern Enumerative Geometry, page 257]] > r ∈ k(X)× > V ⊂ X is a codimension 1 > ξV > a and b in A =X,ξVsuch that r = a/b > ordV(r) = lengthA(A/a) − lengthA(A/b). This is the order of vanishing of r along V > A d-cycle α is said to be rationally equivalent to 0 if it belongs to the subgroup > A∗X = Z∗X/R∗X =nd=0AdX is the Chow group of X, > If X is pure, then ZnX = AnX is freely generated by the classes of the irreducible components of X. Chow groups are covariant for proper morphisms and contravariant for flat morphisms, > f : X → Y be a proper morphism > f∗: A∗X → A∗Y defined on generators by sending a d-cycle class [V ] ∈ AdX to 0 if dim f (V ) < dim V , and to the cycle eV· [f (V )] ∈ AdY if dim V = dim f (V ). Here eVis the degree of the field extension k(f (V )) ⊂ k(V ). > f : X → Y be a flat morphism ### [[An Invitation to Modern Enumerative Geometry.pdf#page=258|An Invitation to Modern Enumerative Geometry, page 258]] > Let π : E → Y be a vector bundle. > though 0∗is hard to describe in general, we informally describe it as ‘intersecting with the zero section’. > Let f : X → Spec k be the structure morphism of a proper k-scheme X. The degree map is by definition the proper pushforward f∗: A∗X → Z. It takes the value 0 on cycle classes of positive dimension. > Let E be a vector bundle of rank r on a scheme X, and let p: P(E) → X be the projective bundle of lines in the fibres of E → X. > LetE(1) be the dual of the tautological line bundleE(−1) ⊂ p∗E on P(E). The Segre classes si(E) can be seen as operators AkX → Ak−iX defined by ### [[An Invitation to Modern Enumerative Geometry.pdf#page=259|An Invitation to Modern Enumerative Geometry, page 259]] > Let E → X be a vector bundle, f : Y → X a proper morphism. One has the projection formula f∗(ci(f∗E) ∩ α) = ci(E) ∩ f∗α, > If f : Y → X is a flat morphism, on the other hand, one has ci(f∗E) ∩ f∗β = f∗(ci(E) ∩ β), β ∈ A > 0 → E → F → G → 0 > Whitney’s formula ct(F ) = ct(E) · ct(G). > The splitting construction says that if E is a vector bundle of rank r on a scheme X, there exists a flat morphism f : Y → X such that the flat pullback f∗: A∗X → A∗Y is injective and f∗E has a filtration > with line bundle quotients Li= Ei/Ei−1 > αi= c1(Li). > Ei−1→ Ei→ Li→ ### [[An Invitation to Modern Enumerative Geometry.pdf#page=260|An Invitation to Modern Enumerative Geometry, page 260]] > f∗ct(E) = > = (1 + α1t)···(1 + αrt). > we can always pretend that E is filtered by 0 = E0⊂ E1⊂ > with line bundle quotients Li, and ct(E) =6ri=1(1 + αit), > where αi= c1(Li). In fact, one should regard (B.3) as a formal expression defining α1,...,αr. These are called the Chern roots of E, and they satisfy the fundamental relation ci(E) = σi(α1,...,αr), i = 0,...,r where σidenotes the i-th symmetric function. For instance, if rk E = r, one would have c0(E) = 1, c1(E) = α1+···+ αr, cr(E) = α1··· αr > The Chern roots of the dual bundle E∗are −α1,..., −αr, so c(E∗) =1≤i≤r(1 − αi). Thus ci(E∗) = (−1)ici(E). > If F is a vector bundle of rank s, the Chern roots of E ⊗ F are αi+ βj > ct(E ⊗ L) =ri=0ct(L)r−ici(E)t ### [[An Invitation to Modern Enumerative Geometry.pdf#page=261|An Invitation to Modern Enumerative Geometry, page 261]] > For the exterior power ∧pE we have ct(∧pE) =6i1<··· The Chern character of a vector bundle E with Chern roots α1,...,αris the expression ch(E) =ri=1exp(αi). > ch(E) = r + c1+12(c21− 2c2) +16(c31− 3c1c2+ 3c3) +··· where ci= ci(E), > ch(F ) = ch(E) + ch(G) for any short exact sequence > ch(E ⊗ E) = ch(E) · ch(E). > the Chern character defines a ring homomorphism ch: K0(X) → A∗X, and if X is a nonsingular projective variety over the complex numbers, ch ⊗ Q is an isomorphism. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=262|An Invitation to Modern Enumerative Geometry, page 262]] > For a vector bundle E with Chern roots α1,...,αr, we set Td(E) =6ri=1αi > Td(E) = 1+12c1+112(c21+c2)+124c1c2+1720(−c41+4c21c2+3c22+c1c3−c4)+··· > Td(F ) = Td(E) · Td(G) holds for every short exact sequence of vector bundles ### [[An Invitation to Modern Enumerative Geometry.pdf#page=264|An Invitation to Modern Enumerative Geometry, page 264]] > Definition B.3.1 (Factorisation) We say that a morphism of schemes f : X → > admits a factorisation if there is a commutative diagram > where i is a closed embedding and π is smooth. > If X, Y are quasiprojective, any morphism X → Y admits a factorisation. A factorisation always exists locally on Y > smooth morphisms and closed immersions are stable under base change and composition. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=265|An Invitation to Modern Enumerative Geometry, page 265]] > TM/YX− NX/M∈ K0(X) > It is called the virtual tangent bundle of f > An lci morphism f : X → Y has a well-defined relative dimension: > r = rk TM/Y− codim(X, M) ∈ Z. For instance, a regular closed immersion of codimension d is an lci morphism of relative dimension −d. > a composition of regular closed immersions X → Y and Y → Z is a regular closed immersion, and there is an exact sequence of vector bundles 0 → NX/Y→ NX/Z→ NY/ZX→ 0. Let f : X → Y be an lci morphism of relative dimension r, > f!: ZkY > σ > Ak+rX. > The first arrow is just flat pullback: it is not a problem to pull back cycles from Yto Msince π is smooth. The arrow σ, called specialisation to the normal cone in ### [[An Invitation to Modern Enumerative Geometry.pdf#page=266|An Invitation to Modern Enumerative Geometry, page 266]] > f!: AkY→ Ak+rX ### [[An Invitation to Modern Enumerative Geometry.pdf#page=268|An Invitation to Modern Enumerative Geometry, page 268]] ### [[An Invitation to Modern Enumerative Geometry.pdf#page=273|An Invitation to Modern Enumerative Geometry, page 273]] > If X → M is a closed immersion of schemes with ideal ℐ ⊂M, the normal cone and the normal sheaf > CX/M= Specn≥0ℐn/ℐn+1 > NX/M= Spec Sym ℐ/ℐ2 > There is always a closed immersion (over X) of CX/Minside NX/M, induced by the natural surjection Sym ℐ/ℐ2n≥0ℐn/ℐ ### [[An Invitation to Modern Enumerative Geometry.pdf#page=274|An Invitation to Modern Enumerative Geometry, page 274]] > Recall that a closed immersion i : X → M is regular of codimension d if for every point x ∈ X there is an affine open neighbourhood i(x) ∈ Spec A ⊂ M such that the ideal I ⊂ A defining X ∩ Spec A → Spec A is generated by a regular sequence of length d. > Let X → M be a closed immersion with ideal ℐ. We have that NX/Mis a vector bundle if and only if ℐ/ℐ2is locally free. > CX/Mis a vector bundle if and only if the natural projection to X is smooth. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=275|An Invitation to Modern Enumerative Geometry, page 275]] > Let ? =n≥0nbe a quasicoherent sheaf of gradedX-algebras. > (†) The canonical map0→Xis an isomorphism,1is coherent and generates ? over0. > A cone over a scheme X is an X-scheme of the form π : Spec ? → X, where ? satisfies (†). > A vector bundle over X is a cone of the form E = Spec Sym ℰ → X, where ℰ is a locally free sheaf of finite rank (and ℰ∗is the sheaf of sections of E). > Let π : C = Spec ? → X > The sheaf ? is recovered from the structural morphism as ? = π∗C > the canonical surjection0 > endows C with a natural X-morphism 0C: X → C, called the zero section of C. The image of the zero section is called the vertex of the cone. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=276|An Invitation to Modern Enumerative Geometry, page 276]] > TX= Spec SymX. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=277|An Invitation to Modern Enumerative Geometry, page 277]] ### [[An Invitation to Modern Enumerative Geometry.pdf#page=278|An Invitation to Modern Enumerative Geometry, page 278]] ### [[An Invitation to Modern Enumerative Geometry.pdf#page=281|An Invitation to Modern Enumerative Geometry, page 281]] > Lf=ℐ/ℐ2α−→ i∗π∈ D[−1,0](X) > unique > A B-stack (or stack over B) is a category fibred in groupoids X → (SchB)étsuch that the isomorphism functors are étale sheaves and all descent data are effective. > An algebraic stack (also called an Artin stack) over B is a B-stack X → (SchB)ét such that the diagonal 1-morphism X → X ×BX is representable (by algebraic spaces), separated and quasicompact, and there is an object U ∈ (SchB)étwith a smooth surjective 1-morphism U → X, called an atlas. > If an atlas U → X of an algebraic stack can be chosen to be étale, then X is called a Deligne–Mumford stack. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=282|An Invitation to Modern Enumerative Geometry, page 282]] > Let f : 풳 → 풴 be a concentrated morphism of algebraic stacks. > Lf∼=fif and only if f is smooth with unramified diagonal. > Moreover, if f = π ◦i is lci, then Lfis perfect of perfect amplitude contained in [−1, 0]. This means that locally it is isomorphic (in the derived category) to a complex of locally free sheaves (of finite rank). > f : 풳 → 풴 and g : 풴 → > Lf∗Lg→ Lg◦f→ Lf > This is called the transitivity triangle. > Y→ Y of an lci morphism f : X → Y > X → M → Y , > f!: AkY→ Ak+rX, ### [[An Invitation to Modern Enumerative Geometry.pdf#page=283|An Invitation to Modern Enumerative Geometry, page 283]] > Recall that for p = k+s, specialisation to the normal cone assigns to a cycle class ### [[An Invitation to Modern Enumerative Geometry.pdf#page=288|An Invitation to Modern Enumerative Geometry, page 288]] > Behrend–Fantechi call the intrinsic normal cone of X. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=292|An Invitation to Modern Enumerative Geometry, page 292]] > Definition C.5.9 An obstruction theory relative to f : X → B is a morphism φ : E → Lfin D(−∞,0](X), such that h0(φ) is an isomorphism and h−1(φ) is a surjection. It is called perfect if E ∈ D[−1,0]coh(X) is of perfect amplitude contained in [−1, 0]. ### [[An Invitation to Modern Enumerative Geometry.pdf#page=295|An Invitation to Modern Enumerative Geometry, page 295]] > It is special to genus 0 maps to convex varieties (like projective spaces, homogeneous spaces, flag varieties) that the expected dimension agrees with the actual dimension. > At a point μ: (C, p1,...,pn) → X, the moduli stack M =g,n(X, β) has tangent space equal to the hyperext group Ext1([μ∗X→C(P )],C) > where we have set P =pi. Obstructions live in the next hyperext group Ext2([μ∗X→C(P )],C). > Although the individual dimension of these spaces is usually impossible to control, their difference is constant and is called the virtual dimension of M. We denote it vdg,n(X, β). > A marked nodal curve (C, p1,...,pn) is stable if and only if it has no infinitesimal automorphisms, that is, Ext0(C(P ),C) = 0. More generally, a map μ: C → X from a marked nodal curve is stable if and only if first the map has no infinitesimal automorphisms, that is, Ext0([μ∗X→C(P )],C) = 0.