Del Pezzo Surfaces Research Articles

On local stability threshold of del Pezzo surfaces

AbstractWe complete the classification of local stability thresholds for smooth del Pezzo surfaces of degree 2. In particular, we show that this number is irrational if and only if there is a unique (‐1)‐curve passing through the point where we are computing the local invariant.

Enumerative geometry of del Pezzo surfaces

We prove an equivalence between the superpotential defined via tropical geometry and Lagrangian Floer theory for special Lagrangian torus fibres in del Pezzo surfaces constructed by Collins-Jacob-Lin [Duke Math. J. 170 (2021), pp. 1291–1375]. We also include some explicit calculations for the projective plane, which confirm some folklore conjectures in this case.

Sporadic and Exceptional

We study the web of correspondences linking the exceptional Lie algebras E8,7,6 and the sporadic simple groups Monster, Baby and the largest Fischer group. This is done via the investigation of classical enumerative problems on del Pezzo surfaces in relation to the cusps of certain subgroups of PSL(2,R) for the relevant McKay-Thompson series in Generalized Moonshine. We also study Conway's sporadic group, as well as its association with the Horrocks-Mumford bundle.

Frobenius actions on Del Pezzo surfaces of degree 2

Frobenius actions on Del Pezzo surfaces of degree 2

Bounding geometrically integral del Pezzo surfaces

Abstract We prove several boundedness statements for geometrically integral normal del Pezzo surfaces X over arbitrary fields. We give an explicit sharp bound on the irregularity if X is canonical or regular. In particular, we show that wild canonical del Pezzo surfaces exist only in characteristic $2$ . As an application, we deduce that canonical del Pezzo surfaces form a bounded family over $\mathbb {Z}$ , generalising work of Tanaka. More generally, we prove the BAB conjecture on the boundedness of $\varepsilon $ -klt del Pezzo surfaces over arbitrary fields of characteristic different from $2, 3$ and $5$ .

Hofer geometry via toric degeneration

The main theme of this paper is to use toric degeneration to study Hofer geometry by producing distinct homogeneous quasimorphisms on the group of Hamiltonian diffeomorphisms. We focus on the (complex n-dimensional) quadric hypersurface and the del Pezzo surfaces, and study two classes of distinguished Lagrangian submanifolds that appear naturally in a toric degeneration, namely the Lagrangian torus which is the monotone fiber of a Lagrangian torus fibration, and the Lagrangian spheres that appear as vanishing cycles. For the quadrics, we prove that the group of Hamiltonian diffeomorphisms admits two distinct homogeneous quasimorphisms and derive some superheaviness results. Along the way, we show that the toric degeneration is compatible with the Biran decomposition. This implies that for n=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n=2$$\\end{document}, the Lagrangian fiber torus (Gelfand–Zeitlin torus) is Hamiltonian isotopic to the Chekanov torus, which answers a question of Y. Kim. We prove analogous results for the del Pezzo surfaces. We also discuss applications to C0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^0$$\\end{document} symplectic topology.

Minimal compactifications of the affine plane with nef canonical divisors

We shall consider minimal analytic compactifications of the affine plane with singularities. In previous work, Kojima and Takahashi proved that any minimal analytic compactification of the affine plane, which has at worse log canonical singularities, is a numerical del Pezzo surface (i.e., a normal complete algebraic surface with the numerically ample anti-canonical divisor) and has only rational singularities. In this article, we show that any minimal analytic compactification of the affine plane with the nef canonical divisor has an irrational singularity.

SymTFTs and duality defects from 6d SCFTs on 4-manifolds

In this work we study particular TQFTs in three dimensions, known as Symmetry Topological Field Theories (or SymTFTs), to identify line defects of two-dimensional CFTs arising from the compactification of 6d (2, 0) SCFTs on 4-manifolds M4. The mapping class group of M4 and the automorphism group of the SymTFT switch between different absolute 2d theories or global variants. Using the combined symmetries, we realize the topological defects in these global variants. Our main example is ℙ1 × ℙ1. For N M5-branes the corresponding 2d theory inherits ℤN 0-form symmetries from the SymTFT. We reproduce the orbifold groupoid for theories with ℤN 0-form symmetries and realize the duality defects at fixed points of the coupling constant under elements of the mapping class group. We also study other Hirzebruch surfaces, del Pezzo surfaces, as well as the connected sum of ℙ1 × ℙ1. We find a rich network of global variants connected via automorphisms and realize more interesting topological defects. Finally, we derive the SymTFT on more general 4-manifolds and provide two examples.

Compactifications of moduli of del Pezzo surfaces via line arrangement and K-stability

Abstract In this paper, we study compactifications of the moduli of smooth del Pezzo surfaces using K-stability and the line arrangement. We construct K-moduli of log del Pezzo pairs with sum of lines as boundary divisors, and prove that for $d=2,3,4$ , these K-moduli of pairs are isomorphic to the K-moduli spaces of del Pezzo surfaces. For $d=1$ , we prove that they are different by exhibiting some walls.

Boundedness of regular del Pezzo surfaces over imperfect fields

For a regular del Pezzo surface X, we prove that |-12KX|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|-12K_X|$$\\end{document} is very ample. Furthermore, we also give an explicit upper bound for the volume KX2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_X^2$$\\end{document} which depends only on [k:kp]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[k: k^p]$$\\end{document} for the base field k. As a consequence, we obtain the boundedness of geometrically integral regular del Pezzo surfaces.

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 $\operatorname {GL}(2,{\mathbb {C}})$.

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.

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.

Moduli of nodal sextic curves via periods of K3 surfaces

In this paper we study the moduli spaces of nodal sextic curves. We realize each irreducible component of the GIT space of sextic curves with given number of nodes as an open subspace of type IV arithmetic quotients. We then focus on the compactifications of the moduli spaces, one side is the geometric (GIT) compactifications, the other side is the Hodge theoretic compactifications such as Looijenga compactifications and Baily-Borel compactifications. The main result is the isomorphism between GIT and Looijenga compactifications. Some examples are closely related to del Pezzo surfaces. We also extend our results to moduli of nodal sextic curves with specified symmetry.

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.

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.

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 $$K^2\leqslant 3$$ .