Del Pezzo Research Articles

Full Exceptional Collections for Anticanonical Log del Pezzo Surfaces

Abstract Motivated by homological mirror symmetry, this paper constructs explicit full exceptional collections for the canonical stacks associated with the series of log del Pezzo surfaces constructed by Johnson and Kollár in [33]. These surfaces have cyclic quotient, non-Gorenstein, singularities. The construction involves both the $\operatorname {GL}(2,{\mathbb {C}})$ McKay correspondence, and the study of the minimal resolutions of the surfaces, which are birational to degree two del Pezzo surfaces. We show that a degree two del Pezzo surface arises in this way if and only if it admits a generalized Eckardt point, and in the course of the paper we classify the blow-ups of ${\mathbb {P}}^{2}$ giving rise to them. Our result on the adjoints of the functor of Ishii–Ueda [32] applies to any finite small subgroup of $\operatorname {GL}(2,{\mathbb {C}})$.

Rank two weak Fano bundles on del Pezzo threefolds of degree five

This paper classifies rank two vector bundles on a del Pezzo threefold [Formula: see text] of degree five whose projectivizations are weak Fano. This classification is then used to determine properties of the moduli spaces of such vector bundles on [Formula: see text], and we determine precisely when the moduli spaces are smooth, irreducible, and fine. We also prove that such a bundle on a del Pezzo threefold of degree one or two splits, and as result gives a classification of weak Fano bundles of rank two on a del Pezzo threefold of Picard rank one.

A Hilbert Irreducibility Theorem for Integral Points on del Pezzo Surfaces

Abstract We prove that the integral points are potentially Zariski dense in the complement of a reduced effective singular anticanonical divisor in a smooth del Pezzo surface, with the exception of ${{\mathbb {P}}}^{2}$ minus three concurrent lines (for which potential density does not hold). This answers positively a question raised by Hassett and Tschinkel and, combined with previous results, completes the proof of the potential density of integral points for complements of anticanonical divisors in smooth del Pezzo surfaces. We then classify the complements that are simply connected and for these we prove that the set of integral points is potentially not thin, as predicted by a conjecture of Corvaja and Zannier.

Classification of Du Val del Pezzo surfaces of Picard rank one in characteristic two and three

In this paper, we classify Du Val del Pezzo surfaces of Picard rank one in characteristic two and three. We also show that if a Du Val del Pezzo surface is Frobenius split, then a general anti-canonical member is smooth. Furthermore, in characteristic two, it is an ordinary elliptic curve.

Characterization of projective varieties beyond varieties of minimal degree and del Pezzo varieties

Varieties of minimal degree and del Pezzo varieties are basic objects in projective algebraic geometry. Those varieties have been characterized and classified for a long time in many aspects. Motivated by the question “which varieties are the most basic and simplest except the above two kinds of varieties in view of geometry and syzygies?”, we give an upper bound of the graded Betti numbers in the quadratic strand and characterize the extremal cases.The extremal varieties of dimension n, codimension e, and degree d are exactly characterized by the following two types:(i)Varieties with d=e+2, depthX=n, and Green-Lazarsfeld index a(X)=0,(ii)Arithmetically Cohen-Macaulay varieties with d=e+3. This is a generalization of G. Castelnuovo, G. Fano, and E. Park's results on the number of quadrics and an extension of the characterizations of varieties of minimal degree and del Pezzo varieties in view of linear syzygies of quadrics due to K. Han and S. Kwak ([6,8,30,16]).In addition, we show that every variety X that belongs to (i) or (ii) is always contained in a unique rational normal scroll Y as a divisor. Also, we describe the divisor class of X in Y.

Non-big Ulrich bundles: the classification on quadrics and the case of small numerical dimension

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.

The 3-dimensional Lyness map and a self-mirror log Calabi–Yau 3-fold

AbstractThe 2-dimensional Lyness map is a 5-periodic birational map of the plane which may famously be resolved to give an automorphism of a log Calabi–Yau surface, given by the complement of an anticanonical pentagon of $$(-1)$$ ( - 1 ) -curves in a del Pezzo surface of degree 5. This surface has many remarkable properties and, in particular, it is mirror to itself. We construct the 3-dimensional big brother of this surface by considering the 3-dimensional Lyness map, which is an 8-periodic birational map. The variety we obtain is a special (non-$${\mathbb {Q}}$$ Q -factorial) affine Fano 3-fold of type $$V_{12}$$ V 12 , and we show that it is a self-mirror log Calabi–Yau 3-fold.

Ulrich bundles on Del Pezzo threefolds

We prove that for any r⩾2 the moduli space of stable Ulrich bundles of rank r and determinant OX(r) on any smooth Fano threefold X of index two is smooth of dimension r2+1 and that the same holds true for even r when the index is four, in which case no odd–rank Ulrich bundles exist. In particular this shows that any such threefold is Ulrich wild. As a preliminary result, we give necessary and sufficient conditions for the existence of Ulrich bundles on any smooth projective threefold in terms of the existence of a curve in the threefold enjoying special properties.

Reflexive polygons and rational elliptic surfaces

In this note we study in detail the geometry of eight rational elliptic surfaces naturally associated to the sixteen reflexive polygons. The elliptic fibrations supported by these surfaces correspond under mirror symmetry to the eight families of smooth del Pezzo surfaces with very ample anticanonical bundle.

Nodal quintic del Pezzo threefolds and their derived categories

We construct a Kawamata type semiorthogondal decomposition for the bounded derived category of coherent sheaves of nodal quintic del Pezzo threefolds, decomposing the bounded derived category into bounded derived categories of finite dimensional algebras. This is achieved by constructing birational maps from nodal quintic del Pezzo threefolds to quadric surface fibrations over the projective line.

Cohomological χ–independence for moduli of one-dimensional sheaves and moduli of Higgs bundles

We prove that the intersection cohomology (together with the perverse and the Hodge filtrations) for the moduli space of one-dimensional semistable sheaves supported in an ample curve class on a toric del Pezzo surface is independent of the Euler characteristic of the sheaves. We also prove an analogous result for the moduli space of semistable Higgs bundles with respect to an effective divisor $D$ of degree $\mathrm{deg}(D)>2g-2$. Our results confirm the cohomological $\chi$-independence conjecture by Bousseau for $\mathbb{P}^2$, and verify Toda's conjecture for Gopakumar-Vafa invariants for certain local curves and local surfaces. For the proof, we combine a generalized version of Ng\^o's support theorem, a dimension estimate for the stacky Hilbert-Chow morphism, and a splitting theorem for the morphism from the moduli stack to the good GIT quotient.

Classifying sections of del Pezzo fibrations, I

We develop a strategy to classify the components of the space of sections of a del Pezzo fibration over \mathbb{P}^1 . In particular, we prove the Movable Bend-and-Break lemma for del Pezzo fibrations. Our approach is motivated by Geometric Manin’s Conjecture and proves upper bounds on the associated counting function. We also give applications to enumerativity of Gromov–Witten invariants and to the study of the Abel–Jacobi map.

Isotopy classes of involutions of del Pezzo surfaces

Let Mn:=CP2#nCP2‾ for 0≤n≤8 be the underlying smooth manifold of a degree 9−n del Pezzo surface. We prove three results about the mapping class group Mod(Mn):=π0(Homeo+(Mn)):1.the classification of, and a structure theorem for, all involutions in Mod(Mn),2.a positive solution to the smooth Nielsen realization problem for involutions of Mn, and3.a purely topological characterization of three remarkable types of involutions on certain Mn coming from birational geometry: de Jonquiéres involutions, Geiser involutions, and Bertini involutions. One main ingredient is the theory of hyperbolic reflection groups.

A Zariski dense exceptional set in Manin’s Conjecture: dimension 2

Recently, Lehmann, Sengupta, and Tanimoto proposed a conjectural construction of the exceptional set in Manin’s Conjecture, which we call the geometric exceptional set. We construct a del Pezzo surface of degree 1 whose geometric exceptional set is Zariski dense. In particular, this provides the first counterexample to the original version of Manin’s Conjecture in dimension 2 in characteristic 0. Assuming the finiteness of Tate-Shafarevich groups of elliptic curves over $${\mathbb Q}$$ with j-invariant 0, we show that there are infinitely many such counterexamples.

On wall-crossing invariance of certain sums of Welschinger numbers

We continue our quest for real enumerative invariants not sensitive to changing the real structure and extend the construction we uncovered previously for counting curves of anti-canonical degree $$\leqslant 2$$ on del Pezzo surfaces with $$K^2=1$$ to curves of any anti-canonical degree and on any del Pezzo surfaces of degree $$K^2\leqslant 3$$ .

Twistors, Quartics, and del Pezzo Fibrations

It has been known that twistor spaces associated to self-dual metrics on compact 4-manifolds are source of interesting examples of non-projective Moishezon threefolds. In this paper we investigate the structure of a variety of new Moishezon twistor spaces. The anti-canonical line bundle on any twistor space admits a canonical half, and we analyze the structure of twistor spaces by using the pluri-half-anti-canonical map from the twistor spaces. Specifically, each of the present twistor spaces is bimeromorphic to a double covering of a scroll of planes over a rational normal curve, and the branch divisor of the double cover is a cut of the scroll by a quartic hypersurface. In particular, the double covering has a pencil of Del Pezzo surfaces of degree two. Correspondingly, the twistor spaces have a pencil of rational surfaces with big anti-canonical class. The base locus of the last pencil is a cycle of rational curves, and it is an anti-canonical curve on smooth members of the pencil. These twistor spaces are naturally classified into four types according to the type of singularities of the branch divisor, or equivalently, those of the Del Pezzo surfaces in the pencil. We also show that the quartic hypersurface satisfies a strong constraint and as a result the defining polynomial of the quartic hypersurface has to be of a specific form. Together with our previous result in Honda (“A new series of compact minitwistor spaces and Moishezon twistor spaces over them”, 2010), the present result completes a classification of Moishezon twistor spaces whose half-anti-canonical system is a pencil. Twistor spaces whose half-anti-canonical system is larger than pencil have been understood for a long time before. In the opposite direction, no example is known of a Moishezon twistor space whose half-anti-canonical system is smaller than a pencil. Twistor spaces which have a similar structure were studied in Honda (“Double solid twistor spaces: the case of arbitrary signature”, 2008 and “Double solid twistor spaces II: General case”, 2015) and they are very special examples among the present twistor spaces.

Admissible subcategories of del Pezzo surfaces

We study admissible subcategories of derived categories of coherent sheaves on del Pezzo surfaces and rational elliptic surfaces. Using a relation between admissible subcategories and anticanonical divisors we prove the following results. First, we classify all admissible subcategories of the projective plane by showing that each is generated by a subcollection of a full exceptional collection. Second, we show that the derived categories of del Pezzo surfaces do not contain any phantom subcategories. This provides first examples of varieties of dimension larger than one that have some nontrivial admissible subcategories, but provably do not contain phantoms. We also prove that any admissible subcategory supported set-theoretically on a smooth (−1)-curve in a surface is generated by some twist of the structure sheaf of that curve.

Positive Ulrich sheaves

AbstractWe provide a criterion for a coherent sheaf to be an Ulrich sheaf in terms of a certain bilinear form on its global sections. When working over the real numbers, we call it a positive Ulrich sheaf if this bilinear form is symmetric or Hermitian and positive-definite. In that case, our result provides a common theoretical framework for several results in real algebraic geometry concerning the existence of algebraic certificates for certain geometric properties. For instance, it implies Hilbert’s theorem on nonnegative ternary quartics, via the geometry of del Pezzo surfaces, and the solution of the Lax conjecture on plane hyperbolic curves due to Helton and Vinnikov.

Rational curves on del Pezzo surfaces in positive characteristic

We study the space of rational curves on del Pezzo surfaces in positive characteristic. For most primes p p we prove the irreducibility of the moduli space of rational curves of a given nef class, extending results of Testa in characteristic 0 0 . We also investigate the principles of Geometric Manin’s Conjecture for weak del Pezzo surfaces. In the course of this investigation, we give examples of weak del Pezzo surfaces defined over F 2 ( t ) \mathbb F_2(t) or F 3 ( t ) \mathbb {F}_{3}(t) such that the exceptional sets in Manin’s Conjecture are Zariski dense.

Mysterious Triality and Rational Homotopy Theory

Mysterious Duality has been discovered by Iqbal, Neitzke, and Vafa (Adv Theor Math Phys 5:769–808, 2002) as a convincing, yet mysterious correspondence between certain symmetry patterns in toroidal compactifications of M-theory and del Pezzo surfaces, both governed by the root system series $$E_k$$ . It turns out that the sequence of del Pezzo surfaces is not the only sequence of objects in mathematics that gives rise to the same $$E_k$$ symmetry pattern. We present a sequence of topological spaces, starting with the four-sphere $$S^4$$ , and then forming its iterated cyclic loop spaces $$\mathscr {L}_c^k S^4$$ , within which we discover the $$E_k$$ symmetry pattern via rational homotopy theory. For this sequence of spaces, the correspondence between its $$E_k$$ symmetry pattern and that of toroidal compactifications of M-theory is no longer a mystery, as each space $$\mathscr {L}_c^k S^4$$ is naturally related to the compactification of M-theory on the k-torus via identification of the equations of motion of $$(11-k)$$ -dimensional supergravity as the defining equations of the Sullivan minimal model of $$\mathscr {L}_c^k S^4$$ . This gives an explicit duality between algebraic topology and physics. Thereby, we extend Iqbal-Neitzke-Vafa’s Mysterious Duality between algebraic geometry and physics into a triality, also involving algebraic topology. Via this triality, duality between physics and mathematics is demystified, and the mystery is transferred to the mathematical realm as duality between algebraic geometry and algebraic topology. Now the question is: Is there an explicit relation between the del Pezzo surfaces $$\mathbb {B}_k$$ and iterated cyclic loop spaces of $$S^4$$ which would explain the common $$E_k$$ symmetry pattern?