3264 and All That: A Second Course in Algebraic Geometry
Abstract
This book can form the basis of a second course in algebraic geometry. As motivation, it takes concrete questions from enumerative geometry and intersection theory, and provides intuition and technique, so that the student develops the ability to solve geometric problems. The authors explain key ideas, including rational equivalence, Chow rings, Schubert calculus and Chern classes, and readers will appreciate the abundant examples, many provided as exercises with solutions available online. Intersection is concerned with the enumeration of solutions of systems of polynomial equations in several variables. It has been an active area of mathematics since the work of Leibniz. Chasles' nineteenth-century calculation that there are 3264 smooth conic plane curves tangent to five given general conics was an important landmark, and was the inspiration behind the title of this book. Such computations were motivation for Poincaré's development of topology, and for many subsequent theories, so that intersection theory is now a central topic of modern mathematics.
... Enumerative geometry [8] deals with the problem of counting the finite number of solutions to special geometric configurations, usually posed in terms of intersection theory and moduli spaces. By a dimensional argument on intersections in the moduli space of projective lines, it is expected that the number of lines contained on a generic complex hypersurface of degree 2n − 3 in projective space CP n is a finite number. ...
... Even though a generating function for C n remains unknown, the linear relations involved in Equation (8) can be inverted to define a generating function of linear combinations of C n that does have a simple closed-form expression. ...
... We review here this procedure deriving a similar general formula for C n , Equation (8), and establishing corollaries 1 and 2. We only use the splitting principle for the top Chern class of the Sym 2n−3 (S * ) bundle over the Grassmannian of projective lines Gr(1, n), and the intersection properties of the basic Schubert cycles σ 1 , σ 1,1 . For all the relevant intersection theory and Schubert calculus via modern algebraic geometry see, e.g., [8,9]. For a review of results on the Fano scheme of lines, see [4]. ...
Two formulas for the classical number $C_n$ of lines on a generic hypersurface of degree $2n-3$ in $\mathbb{CP}^n$ are obtained which differ from the formulas by Dominici, Harris, Libgober, and van der Waerden-Zagier. We review the splitting principle computation by Harris obtaining a similar general closed-form formula in terms of the Catalan numbers and elementary symmetric polynomials. This in turn yields $C_n$ as a linear difference recursion relation of unbounded order. Thus, for the sequence of certain linear combinations of $C_n$, a simple generating function is found. Then, a result from random algebraic geometry by Basu, Lerario, Lundberg, and Peterson, that expresses these classical enumerative invariants as proportional to the Bombieri norm of particular polynomial determinants, yields another combinatorial expansion in terms of certain set compositions and block labeling counting. As an example, we compute this combinatorial interpretation for the cases of 27 lines on a cubic surface and 2875 lines on a quintic threefold. As an application, we reobtain the parity and asymptotic upper bound of the sequence. In an appendix, we generalize the splitting principle calculation to obtain a formula for the number of lines on a generic complete intersection.
... During the Renaissance, different proofs were found by Viète, van Roomen, Gergonne, and Newton (see [10], p. 159). For a pictorial illustration of the theorem, see the cover-page story of the book 3264 and all that [34]. ...
... In 1879 Schubert published the celebrated book Calculus of enumerative geometry [61], which presents the summit of intersection theory in the late 19th century (see [34], p. 2). While developing Chasles's work on conics [13], he demonstrated amazing applications of intersection theory to enumerative geometry, such as 1) the number of conics tangent to 8 general quadrics in space is 4 407 296; ...
... IV) fail to be flag manifolds, but can be constructed by performing a finite number of steps of blow-ups on flag manifolds (see examples in Fulton [38], § 10.4, Eisenbud and Harris [34], Chap. 13, or in [23] for the constructions of the parameter spaces of complete conics and quadrics in CP 3 ). ...
Hilbert's 15th problem called for a rigorous foundation of Schubert calculus, of which a long-standing and challenging part is the Schubert problem of characteristics. In the course of securing a foundation for algebraic geometry, Van der Waerden and Weil attributed this problem to the intersection theory of flag manifolds. This article surveys the background, content, and solution of the problem of characteristics. Our main results are a unified formula for the characteristics and a systematic description of the intersection rings of flag manifolds. We illustrate the effectiveness of the formula and the algorithm by explicit examples. Bibliography: 71 titles.
... Denote by a hyperplane section of S m (1 r , 2). Firstly we recall that the projective variety S m (1 r , 2) is isomorphic to the blowing-up of the projective space ސ m+r along a linear subspace ތ ∼ = ސ r −1 , see for instance [5,Section 9.3.2]. Denote this blowing-up S m (1 r , 2) → ސ m+r by µ and let E be the exceptional divisor. ...
... Linear section V k of the Grassmannian Gr (2,5). The VMRT of the Grassmannian Gr(2, 5) is projectively equivalent to the Segre embedding ސ 1 × ސ 2 ⊆ ސ 5 . ...
... The VMRT of the Grassmannian Gr(2, 5) is projectively equivalent to the Segre embedding ސ 1 × ސ 2 ⊆ ސ 5 . Moreover, for k ≤ 3, there is only one isomorphic class of codimension k linear section V k of Gr (2,5). This implies that the VMRT of V k is projectively equivalent to a general linear section of ސ 1 × ސ 2 with codimension k. ...
... Proposition 2.1 ( [EH16], Prop. 13.12). ...
... The proposition follows from the formula for the Chow ring of a projective bundle (e.g. [EH16], Theorem 9.6). ...
... Proof. By [EH16] Theorem 10.16, deg(Sec n (C)) = n + 1 for all n. The coefficient of E is the generic multiplicity of Sec n (C) along C. We compute this by induction on n. ...
In this paper we examine the cones of effective cycles on blow ups of projective spaces along smooth rational curves. We determine explicitly the cones of divisors and 1- and 2-dimensional cycles on blow ups of rational normal curves, and strengthen these results in cases of low dimension. Central to our results is the geometry of resolutions of the secant varieties of the curves which are blown up, and our computations of their effective cycles may be of independent interest.
... Here we list some main properties of these classes. For more detailed introduction we refer to [EH16]. ...
... We start by recalling Porteous' formula, also called the Giambelli-Thom-Porteous formula. For more details refer to [Ful98] and [EH16]. The following statement is a special case of Theorem 12.4 in [EH16]. ...
... For more details refer to [Ful98] and [EH16]. The following statement is a special case of Theorem 12.4 in [EH16]. ...
In this paper we study a broad class of polynomial optimization problems whose constraints and objective functions exhibit sparsity patterns. We give two characterizations of the number of critical points to these problems, one as a mixed volume and one as an intersection product on a toric variety. As a corollary, we obtain a convex geometric interpretation of polar degrees, a classical invariant of algebraic varieties as well as Euclidean distance degrees. Furthermore, we prove BKK generality of Lagrange systems in many instances. Motivated by our result expressing the algebraic degree of sparse polynomial optimisation problems via Porteus' formula, in the appendix we answer a related question concerning the degree of sparse determinantal varieties.
... The multiplicativity of the total Chern class c(−) with respect to exact sequences [9,Theorem 5.3 (c)] and the fact that c(−) = 1 on trivial bundles then ensure that ...
... Proof This is a simple matter of unwinding the definitions: at (z, (z 1 , · · · , z d−1 )) ∈ E (D) (2)(3)(4)(5)(6)(7)(8)(9)(10) the fiber of q * S β consists of those sections of N ′ that vanish at z and all z i , while that of π * S consists of those that vanish at z. Naturally, the former condition is more restrictive, hence the first claimed inclusion (the other being obvious). As for the ranks, they are the degrees of M = N ′ ⊗ N −1 , N ′ (−z) (for any z ∈ E) and N ′ . ...
... between equidimensional projectivizations, which then restricts to q res : P(π * S/q * S β ) → P β ; (2)(3)(4)(5)(6)(7)(8)(9)(10)(11) note that the domain and codomain of q res have dimensions 2d − 3 and 2d − 2 respectively. This map will give a handle of sorts on the divisor Y = Y E of P β , as we show in Theorem 2.10. ...
Let $\mathcal{E}$ be a rank-2 vector bundle over an elliptic curve $E$, decomposable as a sum of line bundles of degrees $d'>d\ge 2$, and $\mathcal{L}$ the determinant of $\mathcal{E}$. The subspace $L(\mathcal{E})\subset \mathbb{P}^{n-1}\cong \mathbb{P}\mathrm{Ext}^1(\mathcal{L},\mathcal{O}_E)$ consisting of classes of extensions with middle term isomorphic to $\mathcal{E}$ is one of the symplectic leaves of a remarkable Poisson structure on $\mathbb{P}^{n-1}$ defined by Feigin-Odesskii/Polishchuk, and all symplectic leaves arise in this manner, as shown in earlier work that realizes $L(\mathcal{E})$ as the base space of a principal $\mathrm{Aut}(\mathcal{E})$-fibration. Here, we embed $L(\mathcal{E})$ into a larger, projective base space $\widetilde{L}(\mathcal{E})$ of a principal $\mathrm{Aut}(\mathcal{E})$-fibration whose total space consists of sections of $\mathcal{E}$. The embedding realizes $L(\mathcal{E})\subset \widetilde{L}(\mathcal{E})$ as a complement of an anticanonical divisor $Y$ (one of the main results), and we give an explicit description of the normalization of $Y$ as a projective-space bundle over a projective space. For $d=2$ $\widetilde{L}(\mathcal{E})$ is one of the three Hirzebruch surfaces $\Sigma_i$, $i=0,1,2$; we determine which occurs when and hence also the cases when $L(\mathcal{E})$ is affine. Separately, we prove that for $d<\frac n2$ the singular locus of the secant slice $\mathrm{Sec}_{d,z}(E)\subset \mathbb{P}^{n-1}$, the portion of the $d^{th}$ secant variety of $E$ consisting of points lying on spans of $d$-tuples with sum $z\in E$, is precisely $\mathrm{Sec}_{d-2}$. This strengthens result that $L(\mathcal{E})$ is smooth, appearing in prior joint work with R. Kanda and S.P. Smith.
... First, suppose that X is irreducible. By Pieri's formula [EH16,Proposition 4.9], [X * ] · σ 2 = 0. Since σ 2 parametrizes P N −3 's through a point in P N −1 , it follows that the general point of P N −1 is not contained in any of the P N −3 's parametrized by X * . Hence, the P N −3 's parametrized by X * span a subvariety T of dimension N − 2 of P N −1 . ...
... By the degree definition, a general P N 1 section of Z is a general degree d 1 hypersurface in P N 1 . And by [EH16,Theorem 6.34], the scheme of lines in a general degree d 1 hypersurface has codimension d 1 + 1 in G(2, N 1 + 1). Thus, by adding the sections, F 1 (Z) has codimension d 1 + 1 in G(2, N 1 + 1) × P N 2 × · · · × P Nm . ...
... The intersection of [F 1 (Z)] with H N 2 2 · · · H Nm m is the class of lines in a general P N 1 section of Z, which is a degree d 1 hypersurface in P N 1 . So, by [EH16,Proposition 6.4], we can compute the class [F 1 (Z)]: ...
We generalize techniques by Coskun, Riedl, and Yeong, and obtain an almost optimal bound on the degree for the algebraic hyperbolicity of very general hypersurfaces in homogeneous varieties. As examples, we work out the cases of very general hypersurfaces in products of Grassmannians, orthogonal and symplectic Grassmannians, and flag varieties.
... Let L be the line bundle L = O T (b) ⊗ O T (ap * (D j )) over T. From Theorem 9.6 in [21] and Proposition II.7.11 in [20], it follows that ...
... It is worth noting that, in the cases we are interested in, C i = p −1 (P) with P ∈ S(F q δ ), so the maximum number of rational points on C i will be N = #P 1 (F q δ ). Moreover, by Lemma 9.7 in [21], we have that ...
... denotes the closed embedding. As a consequence, by Theorem 9.6 in [21], the Chow ring of T B i is isomorphic to ...
The aim of this article is to give lower bounds on the parameters of algebraic geometric error-correcting codes constructed from projective bundles over Deligne–Lusztig surfaces. The methods based on an intensive use of the intersection theory allow us to extend the codes previously constructed from higher-dimensional varieties, as well as those coming from curves. General bounds are obtained for the case of projective bundles of rank 2 over standard Deligne–Lusztig surfaces, and some explicit examples coming from surfaces of type A2 and 2A4 are given.
... In the final Sect. 6 we extend results from [11] to prove the following Theorem B Let X ⊂ P n+1 be a smooth cubic hypersurface and F(X ) its Fano scheme of lines. Then degree of irrationality of F(X ), i.e., the minimal degree of a dominant, generically finite, rational map to P 2(n−2) , satisfies irr(F(X )) 6. ...
... In fact it is the section induced, under this isomorphism, by f ∈ k[x 0 , . . . , x 5 ] 3 whose vanishing is X (see [6,Proposition 6.4]) and its cohomology class in the Grassmannian is given by c 4 (Sym 3 U * ) which can be computed as follows (see [9,Example 14.7.13]): ...
... Containing a plane is a divisorial condition, so for X outside this locus, we can resolve this map with one blowup F = Bl S F along the surface S. The map φ has been used in various contexts (see, e.g., [2,21]), so it is important to understand its locus of indeterminacy. See also [14,Sections 2,6] for further references and motivation. ...
For a general cubic fourfold X⊂P5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\subset \mathbb {P}^5$$\end{document} with Fano variety F, we compute the Hodge numbers of the locus S⊂F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S\subset F$$\end{document} of lines of second type and the class of the locus V⊂F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V\subset F$$\end{document} of triple lines, using the description of the latter in terms of flag varieties. We also give an upper bound of 6 for the degree of irrationality of the Fano scheme of lines of any smooth cubic hypersurface.
... We make use of some standard results from [13]. Note that c 1 (L ⊗k ) = kc 1 (L ) if L is a line bundle. ...
... Note that there are examples in terms of conformal blocks where the left hand side has rank smaller than k (which is needed for this to be nontrivial). When k = rankE , the left hand side is actually a sort of resultant of E and L ⊗t after taking the dual of one of the two bundles by Proposition 12.1 on p. 428 of [13]). This can be taken with respect to the Chern roots or something more like a "usual" resultant where the variable is the index (i in c i ). ...
We consider the following question: How much of the combinatorial structure determining properties of $\overline{\mathcal{M}_{0, n}}$ is ``intrinsic'' and how much new information do we obtain from using properties specific to this space? Our approach is to study the effect of the $S_n$-action. Apart from being a natural action to consider, it is known that this action does not extend to other wonderful compactifications associated to the $A_{n - 2}$ hyperplane arrangement. We find the differences in intersection patterns of faces on associahedra and permutohedra which characterize the failure to extend to other compactifications and show that this is reflected by most terms of degree $\ge 2$ of the cohomology/Chow ring. Even from a combinatorial perspective, terms of degree 1 are more naturally related to geometric properties. In particular, imposing $S_n$-invariance implies that many of the log concave sequences obtained from degree 1 Hodge--Riemann relations (and all of them for $n \le 2000$) on the Chow ring of $\overline{\mathcal{M}_{0, n}}$ can be restricted to those with a special recursive structure. A conjectural result implies that this is true for all $n$. Elements of these sequences can be expressed as polynomials in quantum Littlewood--Richardson coefficients multiplied by terms such as partition components, factorials, and multinomial coefficients. After dividing by binomial coefficients, polynomials with these numbers as coefficients can be interepreted in terms of volumes or resultants. Finally, we find a connection between the geometry of $\overline{\mathcal{M}_{0, n}}$ and higher degree Hodge--Riemann relations of other rings via Toeplitz matrices.
... We compute the degree by multiplying the classes of these divisors with the class of H ∩ M C in the Chow ring of M C where H is a generic hyperplane of P 2 n −1 . Using Künneth's formula (see [4,Theorem 2.10]), we obtain the Chow rings of M C as follows: ...
... Let i : C X ֒→ M C be the closed immersion of C X in M C . Applying the adjunction formula (see [4,Section 1.4 ...
... contracted curve vs. contraction). Background on divisors and intersection theory may be found in [Ful98,EH16]. ...
Algebraic billiards in a plane curve of degree $d \geq 2$ is a rational correspondence on a surface. The dynamical degree is an algebraic analogue of entropy that measures intersection-theoretic complexity. We compute a lower bound on the dynamical degree of billiards in the generic algebraic curve of degree $d$ over any field of characteristic coprime to $2d$, complementing the upper bound in our previous work. To do this, we specialize to a highly symmetric curve that we call the Fermat hyperbola. Over $\mathbb{C}$, we construct an algebraically stable model for this billiard via an iterated blowup. Over general fields, we compute the growth of a particular big and nef divisor.
... see [20,Proposition 9.13]. By the equivariant splitting principle [24,Theorem 8.6.2], ...
We give a necessary and sufficient condition for the projectivisation of a slope semistable vector bundle to admit constant scalar curvature K\"ahler (cscK) metrics in adiabatic classes, when the base admits a constant scalar curvature metric. More precisely, we introduce a stability condition on vector bundles, which we call adiabatic slope stability, which is a weaker version of K-stability and involves only test configurations arising from subsheaves of the bundle. We prove that, for a simple vector bundle with locally free graded object, adiabatic slope stability is equivalent to the existence of cscK metrics on the projectivisation, which solves a problem that has been open since work of Ross--Thomas. In particular, this shows that the existence of cscK metrics is equivalent to K-stability in this setting. We provide a numerical criterion for the Donaldson-Futaki invariant associated to said test configurations in terms of Chern classes of the vector bundle. This criterion is computable in practice and we present an explicit example satisfying our assumptions which is coming from a vector bundle that does not admit a Hermite-Einstein metric.
... For a partition λ ⊆ r × (n − r) and a positive integer b we have σ λ σ (b) = µ∈λ⊗b σ µ , where λ ⊗ b is the set of partitions in r × (n − r) that can be obtained from the Young diagram of λ by adding b boxes, at most one per column. For a comprehensive reference see [EH16]. ...
The Chow class of the closure of the torus orbit of a point in a Grassmannian only depends on the matroid associated to the point. The Chow class can be extended to a matroid invariant of arbitrary matroids. We call the coefficients appearing in the expansion of the Chow class in the Schubert basis the Schubert coefficients of the matroid. These Schubert coefficients are conjectured by Berget and Fink to be non-negative. We compute the Schubert coefficients for all sparse paving matroids, and confirm their non-negativity.
... Secondly, its singularities are determined by ibid., Lemma 4.8, which says that every singularity of the branch divisor B of f is a point of total ramification, and ibid., Corollary 5.8(ii), which shows that such a total ramification point is a double point of B with one tangent and therefore generally an ordinary cusp. The number of such cusps is then counted in ibid., Lemma 10.1 as 3c 2 (E) in terms of the second Chern class of the Tschirnhausen bundle E. Since this bundle is split in our case, we can compute the Chern classes of E as the elementary symmetric polynomials in the Chern classes of the line bundle O ℙ 2 (−2) so that c 1 (E) = −4L and c 2 (E) = 4L 2 in the Chow ring of ℙ 2 , see [8,Corollary 5.4]. So in summary, the curve B has degree 8, 12 cusps, and genus 9. ...
This is a case study of the algebraic boundary of convex hulls of varieties. We focus on surfaces in fourspace to showcase new geometric phenomena that neither curves nor hypersurfaces exhibit. Our method is a detailed analysis of a general purpose formula by Ranestad and Sturmfels in the case of smooth real algebraic surfaces of low degree (that are rational over the complex numbers). We study both the complex and the real features of the algebraic boundary of Veronese and Del Pezzo surfaces. The main difficulties and the possible approaches to the case of general surfaces are discussed for and complemented by the example of Bordiga surfaces.
... We will recall some basic facts about Schubert calculus in G(k, n). We refer to [2,3] for the proofs and for a complete account on the subject. ...
Denote by $${\mathbb {G}}(k,n)$$ G ( k , n ) the Grassmannian of linear subspaces of dimension k in $${\mathbb {P}}^n$$ P n . We show that if $$\varphi :{\mathbb {G}}(l,n) \rightarrow {\mathbb {G}}(k,n)$$ φ : G ( l , n ) → G ( k , n ) is a nonconstant morphism and $$l \not =0,n-1$$ l ≠ 0 , n - 1 , then $$l=k$$ l = k or $$l=n-k-1$$ l = n - k - 1 and $$\varphi $$ φ is an isomorphism.
... Before giving the statement concerning the structure of the Chow-Witt rings, we discuss the Chow rings of the Grassmannians. This result is very well-known and can be found in the relevant books on intersection theory, such as [11]. ...
... Proof. Recall from [18,Theorem 3.5] that T Gr(k,n) = H om(V, Q) = V ∨ ⊗ Q where Q is the universal quotient bundle satisfying the following short exact sequence ...
We prove a Pieri formula for motivic Chern classes of Schubert cells in the equivariant K-theory of Grassmannians, which is described in terms of ribbon operators on partitions. Our approach is to transform the Schubert calculus over Grassmannians to the calculation in a certain affine Hecke algebra. As a consequence, we derive a Pieri formula for Segre motivic classes of Schubert cells in Grassmannians. We apply the Pieri formulas to establish a relation between motivic Chern classes and Segre motivic classes, extending a well-known relation between the classes of structure sheaves and ideal sheaves. As another application, we find a symmetric power series representative for the class of the dualizing sheaf of a Schubert variety.
... To compute the class of D Res we recall that the degree of a Cayley-Chow form equals the degree of the variety to which it is associated [13, Corollary 2.1], which in our case is equal to the Catalan number C n−2 = 1 [17,Proposition 4.12]. Hence, we have that the divisor classes of D Kosz and D Res are related by ...
We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace K⊆⋀2V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\subseteq \bigwedge ^2 V$$\end{document}, where V is a vector space. Previously Koszul modules of finite length have been used to give a proof of Green’s Conjecture on syzygies of generic canonical curves. We now give applications to effective stabilization of cohomology of thickenings of algebraic varieties, divisors on moduli spaces of curves, enumerative geometry of curves on K3 surfaces and to skew-symmetric degeneracy loci. We also show that the instability of sufficiently positive rank 2 vector bundles on curves is governed by resonance and give a splitting criterion.
... This case is handled by a standard geometric argument, given in, e.g. [10,Proposition 6.15]. ...
In this note we describe explicitly, in terms of Lie theory and cameral data, the covariant (Gauss–Manin) derivative of the Seiberg–Witten differential defined on the weight-one variation of Hodge structures that exists on a Zariski open subset of the base of the Hitchin fibration.
... We identify Pic(X) = A 1 (X), where by Pic(X) we mean the Picard group of X. For a vector bundle E over a smooth projective variety X, c(E) ∈ A * (X) denotes the total Chern class of E. For definitions and results about the Chow ring and Chern classes see [1], [2]. ...
... For a compact Kähler manifold, what effective complex analytic cycles can we "count", and are those "counts" invariant under deformation? In the special case of complex projective manifolds, this is one of the oldest problems in algebraic geometry, dating back at least 150 years to work of Steiner, Chasles, de Jonquiéres and, especially, Schubert [EH16]. Hilbert's search for a rigorous foundation for Schubert's (unpublished) methods for counting cycles eventually led to the invention of the cohomology ring [Kle76]. ...
We study the enumerativity of Gromov-Witten invariants where the domain curve is fixed in moduli and required to pass through the maximum possible number of points. We say a Fano manifold satisfies asymptotic enumerativity if such invariants are enumerative whenever the degree of the curve is sufficiently large. Lian and Pandhari-pande speculate that every Fano manifold satisfies asymptotic enumerativity. We give the first counterexamples, as well as some new examples where asymptotic enumerativity holds. The negative examples include special hypersurfaces of low Fano index and certain projec-tive bundles, and the new positive examples include many Fano threefolds and all smooth hypersurfaces of degree d ≤ (n + 3)/3 in P n .
... Because the homogeneous coordinate ring of the Grassmannian is Cohen-Macaulay, we can relate this to the degree of the generators of the canonical module. The canonical module of the Grassmannian G(m, n) is O G (−n) in the Plücker embedding (see for example [EH,Proposition 5.25]). Thus modulo a general sequence of ℓ(I) = m(n − m) + 1 linear forms, the socle is in degree ℓ(I) − n, and the reduction number is thus r(I) = ℓ(I) − n. □ It is interesting to ask when an ideal of maximal minors has an (ℓ − 1)-residual intersection, so that part (2) of Theorem 2.1 applies. ...
If I I is an ideal in a Gorenstein ring S S , and S / I S/I is Cohen-Macaulay, then the same is true for any linked ideal I ′ I’ ; but such statements hold for residual intersections of higher codimension only under restrictive hypotheses, not satisfied even by ideals as simple as the ideal L n L_{n} of minors of a generic 2 × n 2 \times n matrix when n > 3 n>3 .
In this paper we initiate the study of a different sort of Cohen-Macaulay property that holds for certain general residual intersections of the maximal (interesting) codimension, one less than the analytic spread of I I . For example, suppose that K K is the residual intersection of L n L_{n} by 2 n − 4 2n-4 general quadratic forms in L n L_{n} . In this situation we analyze S / K S/K and show that I n − 3 ( S / K ) I^{n-3}(S/K) is a self-dual maximal Cohen-Macaulay S / K S/K -module with linear free resolution over S S .
The technical heart of the paper is a result about ideals of analytic spread 1 whose high powers are linearly presented.
... We recall the following well-known fact (see for example [6,Prop. 10.2]). ...
On any smooth n-dimensional variety we give a pretty precise picture of rank r Ulrich vector bundles with numerical dimension at most n2+r-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{n}{2}+r-1$$\end{document}. Also, we classify non-big Ulrich vector bundles on quadrics and on the Del Pezzo fourfold of degree 6.
... We will recall some basic facts about Schubert calculus in G(k, n). We refer to [1] and [3] for the proofs and for a complete account on the subject. ...
Denote by $\mathbb G(k,n)$ the Grassmannian of linear subspaces of dimension $k$ in $\mathbb P^n$. We show that, if $\varphi:\mathbb G(l,n) \to \mathbb G(k,n)$ is a non constant morphism and $l \not=0,n-1$ then $l=k$ or $l=n-k-1$ and $\varphi$ is an isomorphism.
... Let us denote by 1 ∈ CH 1 ( ), 2 ∈ CH 2 ( ) the image under of the first (resp, second) Chern class of the tautological bundle on the Grassmannian Gr(2, 5). Recall that CH * (Gr(2, 5)) is generated as a ℚ-algebra by the first and second Chern classes of [16]. Then, the ℚ-algebra ⟨Im ⊗ Im , Δ ⟩ is generated by Δ and the pull-backs of 1 and 2 with respect to the two projections ∶ × → , = 1, 2. By abuse of notation, in the rest of the proof, we will denote by also the pull-back of via anyone of the two projections. ...
Gushel–Mukai (GM) sixfolds are an important class of so‐called Fano‐K3 varieties. In this paper, we show that they admit a multiplicative Chow–Künneth decomposition modulo algebraic equivalence and that they have the Franchetta property. As side results, we show that double Eisenbud‐Popescu‐Walter (EPW) sextics and cubes have the Franchetta property, modulo algebraic equivalence, and some vanishing results for the Chow ring of GM sixfolds.
... Our convention for the projectivization of a vector bundle E on a variety X is that P X (E) = Proj X Sym • (E ∨ ), which agrees with[EH16] but is the opposite of[Har77,Laz04b]. ...
Matroids are combinatorial abstractions of independence, a ubiquitous notion that pervades many branches of mathematics. June Huh and his collaborators recently made spectacular breakthroughs by developing a Hodge theory of matroids that resolved several long-standing conjectures in matroid theory. We survey the main results in this development and ideas behind them.
... Since the Chow ring of Gr(2, 5) is generated by the Plücker hyperplane and c 2 of the tautological bundle [12], this gives the expression for the generically defined cycles. For the injectivity, this is true for trivial reasons: Y is a Fano fourfold and hence A 4 hom (Y ) = 0, and so (as can be se seen from the Bloch-Srinivas argument [4]) we have B * hom (Y ) = 0. Lemma 2.17. ...
We prove that the Beauville-Voisin conjecture is true for any double EPW sextic, i.e. the subalgebra of the Chow ring generated by divisors and Chern classes of the tangent bundle injects into cohomology.
... We use the definition of P(E) := Proj(Sym E) as found in [7]. It is worthwhile noticing, to avoid any confusion, that in some texts like [11], some authors define a projective bundle: "P(E) = Proj(Sym E ∨ )" that would be, in our notation, P(E ∨ ). ...
We solve the Ein–Lazarsfeld–Mustopa conjecture for the blow up of a projective space along a linear subspace. More precisely, let X be the blow up of $${\mathbb {P}}^n$$ P n at a linear subspace and let L be any ample line bundle on X . We show that the syzygy bundle $$M_{L}$$ M L defined as the kernel of the evalution map $$H^{0}(X,L)\otimes {\mathcal {O}}_{X}\rightarrow L$$ H 0 ( X , L ) ⊗ O X → L is L -stable. In the last part of this note we focus on the rigidness of $$M_{L}$$ M L to study the local shape of the moduli space around the point $$[M_{L}]$$ [ M L ] .
Between 1873 and 1876, Felix Klein published a series of papers that he later placed under the rubric anschauliche Geometrie in the second volume of his collected works (1922). The present study attempts not only to follow the course of this work, but also to place it in a larger historical context. Methodologically, Klein’s approach had roots in Poncelet’s principle of continuity, though the more immediate influences on him came from his teachers, Plücker and Clebsch. In the 1860s, Clebsch reworked some of the central ideas in Riemann’s theory of Abelian functions to obtain complicated results for systems of algebraic curves, most published earlier by Hesse and Steiner. These findings played a major role in enumerative geometry, whereas Plücker’s work had a strongly qualitative character that imbued Klein’s early studies. A leitmotif in these works can be seen in the interplay between real curves and surfaces as reflected by their transformational properties. During the early 1870s, Klein and Zeuthen began to explore the possibility of deriving all possible forms for real cubic surfaces as well as quartic curves. They did so using continuity methods reminiscent of Poncelet’s earlier approach. Both authors also relied on visual arguments, which Klein would later advance under the banner of intuitive geometry (anschauliche Geometrie).
Recently, H. Dao and R. Nair gave a combinatorial description of simplicial complexes \(\Delta \) such that the squarefree reduction of the Stanley–Reisner ideal of \(\Delta \) has the WLP in degree 1 and characteristic zero. In this paper, we apply the connections between analytic spread of equigenerated monomial ideals, mixed multiplicities and birational monomial maps to give a sufficient and necessary condition for the squarefree reduction \(A(\Delta )\) to satisfy the WLP in degree i and characteristic zero in terms of mixed multiplicities of monomial ideals that contain combinatorial information of \(\Delta \), we call them incidence ideals. As a consequence, we give an upper bound to the possible failures of the WLP of \(A(\Delta )\) in degree i in positive characteristics in terms of mixed multiplicities. Moreover, we extend Dao and Nair’s criterion to arbitrary monomial ideals in positive odd characteristics.
We compute the Chow ring of the moduli stack of planar nodal curves of fixed degree and express it in terms of tautological classes. Along the way, we extend Vial's results on Chow groups of Brauer-Severi varieties to $G$-equivariant settings.
We find sharp upper bounds on the order of the automorphism group of a hypersurface in complex projective space in every dimension and degree. In each case, we prove that the hypersurface realizing the upper bound is unique up to isomorphism and provide explicit generators for the automorphism group. Similar results were recently proven independently by S. Yang, X. Yu, and Z. Zhu using different methods.
We solve the problem of Apollonius by applying the Lorentz transformation within the frame- work of special relativity to go to a frame of simultaneity, where the solutions are the circum- center. We transform back to get the solutions in a general frame. We show that the solutions in a general frame are the foci of the ellipse going through the centers of the given 3 circles.
We prove that a smooth projective variety $X$ of dimension $n$ with strictly nef third, fourth, or $(n-1)$-th exterior power of the tangent bundle is a Fano variety. Moreover, in the first two cases, we provide a classification for $X$ under the assumption that $\rho (X)\ne 1$.
We consider the bit complexity of computing Chow forms of projective varieties defined over integers and their generalization to multiprojective spaces. We develop a deterministic algorithm using resultants and obtain a single exponential complexity upper bound. Earlier computational results for Chow forms were in the arithmetic complexity model; thus, our result represents the first bit complexity bound. We also extend our algorithm to Hurwitz forms in projective space and we explore connections between multiprojective Hurwitz forms and matroid theory. The motivation for our work comes from incidence geometry where intriguing computational algebra problems remain open.
Consider a compact group $G$ acting on a real or complex Banach Lie group $U$, by automorphisms in the relevant category, and leaving a central subgroup $K\le U$ invariant. We define the spaces ${}_KZ^n(G,U)$ of $K$-relative continuous cocycles as those maps ${G^n\to U}$ whose coboundary is a $K$-valued $(n+1)$-cocycle; this applies to possibly non-abelian $U$, in which case $n=1$. We show that the ${}_KZ^n(G,U)$ are analytic submanifolds of the spaces $C(G^n,U)$ of continuous maps $G^n\to U$ and that they decompose as disjoint unions of fiber bundles over manifolds of $K$-valued cocycles. Applications include: (a) the fact that ${Z^n(G,U)\subset C(G^n,U)}$ is an analytic submanifold and its orbits under the adjoint of the group of $U$-valued $(n-1)$-cochains are open; (b) hence the cohomology spaces $H^n(G,U)$ are discrete; (c) for unital $C^*$-algebras $A$ and $B$ with $A$ finite-dimensional the space of morphisms $A\to B$ is an analytic manifold and nearby morphisms are conjugate under the unitary group $U(B)$; (d) the same goes for $A$ and $B$ Banach, with $A$ finite-dimensional and semisimple; (e) and for spaces of projective representations of compact groups in arbitrary $C^*$ algebras (the last recovering a result of Martin's).
For the symplectic Grassmannian SpG(2,2n) of 2-dimensional isotropic subspaces in a 2n-dimensional vector space over an algebraically closed field of characteristic zero endowed with a symplectic form and with the natural action of an n-dimensional torus T on it, we characterize its irreducible T-invariant subvarieties. This characterization is in terms of symplectic Coxeter matroids, and we use this result to give a complete characterization of the symplectic matroids of rank 2 which are representable over C.
We prove that the invariant locus of the involution associated with a general double Eisenbud-Popescu-Walter (EPW) sextic is a constant cycle surface and introduce a filtration on CH1 of a Gushel-Mukai fourfold. We verify the sheaf/cycle correspondence for sheaves supported on low-degree rational curves, parallel to the case of cubic fourfolds of Shen-Yin’s work.
We establish a general computational scheme designed for a systematic computation of characteristic classes of singular complex algebraic varieties that satisfy a Gysin axiom in a transverse setup. This scheme is explicitly geometric and of a recursive nature terminating on genera of explicit characteristic subvarieties that we construct. It enables us, for example, to apply intersection theory of Schubert varieties to obtain a uniqueness result for such characteristic classes in the homology of an ambient Grassmannian. Our framework applies in particular to the Goresky–MacPherson ‐class by virtue of the Gysin restriction formula obtained by the first author in previous work. We illustrate our approach for a systematic computation of the ‐class in terms of normally nonsingular expansions in examples of singular Schubert varieties that do not satisfy Poincaré duality over the rationals.
We use the geometry of the stellahedral toric variety to study matroids. We identify the valuative group of matroids with the cohomology ring of the stellahedral toric variety and show that valuative, homological and numerical equivalence relations for matroids coincide. We establish a new log-concavity result for the Tutte polynomial of a matroid, answering a question of Wagner and Shapiro–Smirnov–Vaintrob on Postnikov–Shapiro algebras, and calculate the Chern–Schwartz–MacPherson classes of matroid Schubert cells. The central construction is the ‘augmented tautological classes of matroids’, modeled after certain toric vector bundles on the stellahedral toric variety.
The aim of this paper is to determine a bound of the dimension of an irreducible component of the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces. Let X be a nonsingular irreducible complex surface, and let E be a vector bundle of rank n on X . We use the m -elementary transformation of E at a point $x \in X$ to show that there exists an embedding from the Grassmannian variety $\mathbb{G}(E_x,m)$ into the moduli space of torsion-free sheaves $\mathfrak{M}_{X,H}(n;\,c_1,c_2+m)$ which induces an injective morphism from $X \times M_{X,H}(n;\,c_1,c_2)$ to $Hilb_{\, \mathfrak{M}_{X,H}(n;\,c_1,c_2+m)}$ .
In this paper, we consider products of \(\psi \) and \(\omega \) classes on \(\overline{M}_{0,n+3}\). For each product, we construct a flat family of subschemes of \(\overline{M}_{0,n+3}\) whose general fiber is a complete intersection representing the product, and whose special fiber is a generically reduced union of boundary strata. Our construction is built up inductively as a sequence of one-parameter degenerations, using an explicit parametrized collection of hyperplane sections. Combinatorially, our construction expresses each product as a positive, multiplicity-free sum of classes of boundary strata. These are given by a combinatorial algorithm on trees we call slide labeling. As a corollary, we obtain a positive combinatorial formula for the \(\kappa \) classes in terms of boundary strata. For degree-n products of \(\omega \) classes, the special fiber is a finite reduced union of (boundary) points, and its cardinality is one of the multidegrees of the corresponding embedding \(\Omega _n:\overline{M}_{0,n+3}\rightarrow \mathbb {P}^1\times \cdots \times \mathbb {P}^n\). In the case of the product \(\omega _1\cdots \omega _n\), these points exhibit a connection to permutation pattern avoidance. Finally, we show that in certain cases, a prior interpretation of the multidegrees via tournaments can also be obtained by degenerations.
We use recent duality results of Eisenbud and Ulrich to give tools to study quadratically enriched residual intersections when there is no excess bundle. We use this to prove a formula for the Witt-valued Euler number of an almost complete intersection. We give example computations of quadratically enriched excess and residual intersections.
A chopped ideal is obtained from a homogeneous ideal by considering only the generators of a fixed degree. We investigate cases in which the chopped ideal defines the same finite set of points as the original one-dimensional ideal. The complexity of computing these points from the chopped ideal is governed by the Hilbert function and regularity. We conjecture values for these invariants and prove them in many cases. We show that our conjecture is of practical relevance for symmetric tensor decomposition.
We compute the Chow groups and the Fulton-MacPherson operational Chow cohomology ring for a class of singular rational varieties including toric varieties. The computation is closely related to the weight filtration on the ordinary cohomology of these varieties. We use the computation to answer one of the open problems about operational Chow cohomology: it does not have a natural map to ordinary cohomology.
Hassett and Tschinkel gave counterexamples to the integral Hodge conjecture among 3-folds over a number field. We work out their method in detail, showing that essentially all known counterexamples to the integral Hodge conjecture over the complex numbers can be made to work over a number field.