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We study the categorical Torelli theorem for smooth (weighted) hypersurfaces in (weighted) projective spaces via the Hochschild--Serre algebra of its Kuznetsov component. In the first part of the paper, we show that a natural graded subalgebra of the Hochschild--Serre algebra of the Kuznetsov component of a degree $d$ weighted hypersurface in $\mathbb{P}(a_0,\ldots,a_n)$ reconstructs the graded subalgebra of the Jacobian ring generated by the degree $t:=\mathrm{gcd}(d,\Sigma_{i=0}^na_i)$ piece under mild assumptions. Using results of Donagi and Cox--Green, this gives a categorical Torelli theorem for most smooth hypersurfaces $Y$ of degree $d \le n$ in $\mathbb{P}^n$ such that $d$ does not divide $n+1$ (the exception being the cases of the form $(d,n) = (4, 4k + 2)$, for which a result of Voisin lets us deduce a generic categorical Torelli theorem when $k \ge 150$). Next, we show that the Jacobian ring of the Veronese double cone can be reconstructed from its graded subalgebra of even degree, thus proving a categorical Torelli theorem for the Veronese double cone. In the second part, we rebuild the infinitesimal Variation of Hodge structures of a series of (weighted) hypersurfaces from their Kuznetsov components via the Hochschild--Serre algebra. As a result, we prove categorical Torelli theorems for two classes of (weighted) hypersurfaces: $(1):$ Generalized Veronese double cone; $(2):$ Certain $k$-sheeted covering of $\mathbb{P}^n$, when they are generic. Then, we prove a refined categorical Torelli theorem for a Fano variety whose Kuznetsov component is a Calabi--Yau category of dimension $2m+1$. Finally, we prove the actual categorical Torelli theorem for generalized Veronese double cone and $k$-sheeted covering of $\mathbb{P}^n$., Comment: 28 pages, comments are very welcome

Since language models (LMs) now outperform average humans on many challenging tasks, it has become increasingly difficult to develop challenging, high-quality, and realistic evaluations. We address this issue by examining LMs' capabilities to generate code for solving real scientific research problems. Incorporating input from scientists and AI researchers in 16 diverse natural science sub-fields, including mathematics, physics, chemistry, biology, and materials science, we created a scientist-curated coding benchmark, SciCode. The problems in SciCode naturally factorize into multiple subproblems, each involving knowledge recall, reasoning, and code synthesis. In total, SciCode contains 338 subproblems decomposed from 80 challenging main problems. It offers optional descriptions specifying useful scientific background information and scientist-annotated gold-standard solutions and test cases for evaluation. Claude3.5-Sonnet, the best-performing model among those tested, can solve only 4.6% of the problems in the most realistic setting. We believe that SciCode demonstrates both contemporary LMs' progress towards becoming helpful scientific assistants and sheds light on the development and evaluation of scientific AI in the future., Comment: 25 pages, 9 figures, 7 tables

Let $Y$ be a cubic threefold with a non-Eckardt type involution $\tau$. Our first main result is that the $\tau$-equivariant category of the Kuznetsov component $\mathcal{K}u_{\mathbb{Z}_2}(Y)$ determines the isomorphism class of $Y$ for general $(Y,\tau)$. We shall prove this categorical Torelli theorem via three approaches: a noncommutative Hodge theoretical one (using a generalization of the intermediate Jacobian construction due to Alexander Perry), a Bridgeland moduli theoretical one (using equivariant stability conditions), and a Chow theoretical one (using some techniques in [kuznetsovnonclodedfield2021]).The remaining part of the paper is devoted to proving an equivariant infinitesimal categorical Torelli for non-Eckardt cubic threefolds $(Y,\tau)$. To accomplish it, we prove a compatibility theorem on the algebra structures of the Hochschild cohomology of the bounded derived category $D^b(X)$ of a smooth projective variety $X$ and on the Hochschild cohomology of a semi-orthogonal component of $D^b(X)$. Another key ingredient is a generalization of a result in [macri2009infinitesima] which shows that the twisted Hochschild-Kostant-Rosenberg isomorphism is compatible with the actions on the Hochschild cohomology and on the singular cohomology induced by an automorphism of $X$. In appendix, we prove an equivariant categorical Torelli theorem for arbitrary cubic threefold with a geometric involution under a natural assumption., Comment: Add an appendix on an equivariant categorical Torelli theorem for all cubic threefold with a geometric involution under a natural assumption

Let $X$ be a very general Gushel-Mukai (GM) variety of dimension $n\geq 4$, and let $Y$ be a smooth hyperplane section. There are natural pull-back and push-forward functors between the semi-orthogonal components (known as the Kuznetsov components) of the derived categories of $X$ and $Y$. In the first part of the paper, we prove that the Bridgeland stability of objects is preserved by both pull-back and push-forward functors. We then explore various applications of this result, such as constructing an 8-dimensional smooth family of Lagrangian subvarieties for each moduli space of stable objects in the Kuznetsov component of a general GM fourfold, and proving the projectivity of the moduli spaces of semistable objects of any class in the Kuznetsov component of a general GM threefold, as conjectured by Perry, Pertusi, and Zhao. In the second part, we study in detail the double EPW cube, a 6-dimensional hyperk\"ahler manifold associated with a very general GM fourfold $X$, through the Bridgeland moduli space, and show that it is the maximal rationally connected (MRC) quotient of the Hilbert scheme of twisted cubics on $X$. We also prove that it admits a covering by Lagrangian subvarieties, constructed from the Hilbert schemes of twisted cubics on GM threefolds., Comment: 62 pages. Comments are very welcome! Ver2: results are generalized to general cases

Solely relying on test passing to evaluate Large Language Models (LLMs) for code synthesis may result in unfair assessment or promoting models with data leakage. As an alternative, we introduce CodeMind, a framework designed to gauge the code reasoning abilities of LLMs. CodeMind currently supports three code reasoning tasks: Independent Execution Reasoning (IER), Dependent Execution Reasoning (DER), and Specification Reasoning (SR). The first two evaluate models to predict the execution output of an arbitrary code or code the model could correctly synthesize. The third one evaluates the extent to which LLMs implement the specified expected behavior. Our extensive evaluation of nine LLMs across five benchmarks in two different programming languages using CodeMind shows that LLMs fairly follow control flow constructs and, in general, explain how inputs evolve to output, specifically for simple programs and the ones they can correctly synthesize. However, their performance drops for code with higher complexity, non-trivial logical and arithmetic operators, non-primitive types, and API calls. Furthermore, we observe that, while correlated, specification reasoning (essential for code synthesis) does not imply execution reasoning (essential for broader programming tasks such as testing and debugging): ranking LLMs based on test passing can be different compared to code reasoning.

Let $Q_i(i=1,2)$ be $2g$ dimensional quadrics in $\mathbb{P}^{2g+1}$ and let $Y$ be the smooth intersection $Q_1\cap Q_2$. We associate the linear subspace in $Y$ with vector bundles on the hyperelliptic curve $C$ of genus $g$ by the left adjoint functor of $\Phi:D^b(C)\rightarrow D^b(Y)$. As an application, we give a different proof of the classification of line bundles and stable bundles of rank $2$ on hyperelliptic curves given by Desale and Ramanan. When $g=3$, we show that the projection functor induces a closed embedding $\alpha:Y\rightarrow SU^s_C(4,h)$ into the moduli space of stable bundles on $C$ of rank $4$ of fixed determinant., Comment: 15 pages

Let $Y$ be a smooth quartic double solid regarded as a degree 4 hypersurface of the weighted projective space $\mathbb{P}(1,1,1,1,2)$. We study the multiplication of Hochschild-Serre algebra of its Kuznetsov component $\mathcal{K}u(Y)$, via matrix factorization. As an application, we give a new disproof of Kuznetsov's Fano threefold conjecture., Comment: 13 pages, add more details in the proof of Theorem 3.1, correct typos, remove the appendix which is not relevant to the main topic. Final version, to appear in Math.Z

Let $X$ be a smooth Fano variety. We attach a bi-graded associative algebra $\mathrm{HS}(\mathcal{K}u(X))=\bigoplus_{i,j\in \mathbb{Z}} \mathrm{Hom}(\mathrm{Id},S_{\mathcal{K}u(X)}^{i}[j])$ to the Kuznetsov component $\mathcal{K}u(X)$ whenever it is defined. Then we construct a natural sub-algebra of $\mathrm{HS}(\mathcal{K}u(X))$ when $X$ is a Fano hypersurface and establish its relation with Jacobian ring $\mathrm{Jac}(X)$. As an application, we prove a categorical Torelli theorem for Fano hypersurface $X\subset\mathbb{P}^n(n\geq 2)$ of degree $d$ if $\mathrm{gcd}(n+1,d)=1.$ In addition, we give a new proof of the $[Pir22, Theorem 1.2]$ using a similar idea., Comment: 13 pages, final version. To appear in Math Ann

Neural networks have in recent years shown promise for helping software engineers write programs and even formally verify them. While semantic information plays a crucial part in these processes, it remains unclear to what degree popular neural architectures like transformers are capable of modeling that information. This paper examines the behavior of neural networks learning algorithms relevant to programs and formal verification proofs through the lens of mechanistic interpretability, focusing in particular on structural recursion. Structural recursion is at the heart of tasks on which symbolic tools currently outperform neural models, like inferring semantic relations between datatypes and emulating program behavior. We evaluate the ability of transformer models to learn to emulate the behavior of structurally recursive functions from input-output examples. Our evaluation includes empirical and conceptual analyses of the limitations and capabilities of transformer models in approximating these functions, as well as reconstructions of the ``shortcut" algorithms the model learns. By reconstructing these algorithms, we are able to correctly predict 91 percent of failure cases for one of the approximated functions. Our work provides a new foundation for understanding the behavior of neural networks that fail to solve the very tasks they are trained for.

Large language models have the potential to simplify formal theorem proving and make it more accessible. But how to get the most out of these models is still an open question. To answer this question, we take a step back and explore the failure cases of these models using common prompting-based techniques. Our talk will discuss these failure cases and what they can teach us about how to get more out of these models.

Deep Learning (DL) library bugs affect downstream DL applications, emphasizing the need for reliable systems. Generating valid input programs for fuzzing DL libraries is challenging due to the need for satisfying both language syntax/semantics and constraints for constructing valid computational graphs. Recently, the TitanFuzz work demonstrates that modern Large Language Models (LLMs) can be directly leveraged to implicitly learn all the constraints to generate valid DL programs for fuzzing. However, LLMs tend to generate ordinary programs following similar patterns seen in their massive training corpora, while fuzzing favors unusual inputs that cover edge cases or are unlikely to be manually produced. To fill this gap, this paper proposes FuzzGPT, the first technique to prime LLMs to synthesize unusual programs for fuzzing. FuzzGPT is built on the well-known hypothesis that historical bug-triggering programs may include rare/valuable code ingredients important for bug finding. Traditional techniques leveraging such historical information require intensive human efforts to design dedicated generators and ensure the validity of generated programs. FuzzGPT demonstrates that this process can be fully automated via the intrinsic capabilities of LLMs (including fine-tuning and in-context learning), while being generalizable and applicable to challenging domains. While FuzzGPT can be applied with different LLMs, this paper focuses on the powerful GPT-style models: Codex and CodeGen. Moreover, FuzzGPT also shows the potential of directly leveraging the instruct-following capability of the recent ChatGPT for effective fuzzing. Evaluation on two popular DL libraries (PyTorch and TensorFlow) shows that FuzzGPT can substantially outperform TitanFuzz, detecting 76 bugs, with 49 already confirmed as previously unknown bugs, including 11 high-priority bugs or security vulnerabilities.

Let $Y_d$ be a del Pezzo threefold of Picard rank one and degree $d\geq 2$. In this paper, we apply two different viewpoints to study $Y_d$ via a particular admissible subcategory of its bounded derived category, called the Kuznetsov component: (i) Brill-Noether reconstruction. We show that $Y_d$ can be uniquely recovered as a Brill-Noether locus of Bridgeland stable objects in its Kuznetsov component. (ii) Exact equivalences. We prove that, up to composing with an explicit auto-equivalence, any Fourier-Mukai type equivalence of Kuznetsov components of two del Pezzo threefolds of degree $2\leq d\leq 4$ can be lifted to an equivalence of their bounded derived categories. As a result, we obtain a complete description of the group of Fourier-Mukai type auto-equivalences of the Kuznetsov component of $Y_d$. We also describe the group of Fourier-Mukai type auto-equivalences of Kuznetsov components of index one prime Fano threefolds $X_{2g-2}$ of genus $g=6$ and $8$. As an application, first we identify the group of automorphisms of $X_{14}$ and its associated $Y_3$. Then we give a new disproof of Kuznetsov's Fano threefold conjecture by assuming Gushel-Mukai threefolds are general. In an appendix, we classify instanton sheaves on quartic double solids, generalizing a result of Druel., Comment: 35 pages, added results on index one prime Fano threefolds and the application to Kuznetsov's Fano threefold conjecture. Comments are very welcome!

We show by a uniform argument that every index one prime Fano threefold $X$ of genus $g\geq 6$ can be reconstructed as a Brill--Noether locus inside a Bridgeland moduli space of stable objects in the Kuznetsov component $\mathcal{K}u(X)$. As an application, we prove refined categorical Torelli theorems for $X$ and compute the fiber of the period map for each Fano threefold of genus $g\geq 7$ in terms of a certain gluing object associated with the subcategory $\langle \mathcal{O}_X \rangle^{\perp}$. This unifies results of Mukai, Brambilla-Faenzi, Debarre-Iliev-Manivel, Faenzi-Verra, Iliev-Markushevich-Tikhomirov and Kuznetsov., Comment: 45 pages. Comments are very welcome!

Few-shot learning is a challenging task that requires language models to generalize from limited examples. Large language models like GPT-3 and PaLM have made impressive progress in this area, but they still face difficulties in reasoning tasks such as GSM8K, a benchmark for arithmetic problems. To improve their reasoning skills, previous work has proposed to guide the language model with prompts that elicit a series of reasoning steps before giving the final answer, achieving a significant improvement on GSM8K from 17.9% to 58.1% in problem-solving rate. In this paper, we present DIVERSE (Diverse Verifier on Reasoning Step), a novel approach that further enhances the reasoning capability of language models. DIVERSE has three main components: first, it generates diverse prompts to explore different reasoning paths for the same question; second, it uses a verifier to filter out incorrect answers based on a weighted voting scheme; and third, it verifies each reasoning step individually instead of the whole chain. We evaluate DIVERSE on the latest language model code-davinci-002 and show that it achieves new state-of-the-art results on six of eight reasoning benchmarks (e.g., GSM8K 74.4% to 83.2%).

We propose two conjectures on a moduli theoretic approach to constructing Lagrangian subvarieties of hyperk\"ahler varieties arising from the Kuznetsov components of cubic fourfolds or Gushel--Mukai fourfolds. Then we verify the conjectures in several cases, recovering classical examples. As a corollary, we confirm a conjecture of O'Grady in several instances on the existence of Lagrangian covering families for hyperk\"ahler varieties., Comment: 20 pages. Comments are welcome!

Let $X$ be a smooth Fano variety and $\mathcal{K}u(X)$ the Kuznetsov component. Torelli theorems for $\mathcal{K}u(X)$ says that it is uniquely determined by a polarized abelian variety attached to it. An infinitesimal Torelli theorem for $X$ says that the differential of the period map is injective. A categorical variant of infinitesimal Torelli theorem for $X$ says that the morphism $H^1(X,T_X)\xrightarrow{\eta} HH^2(\mathcal{K}u(X))$ is injective. In the present article, we use the machinery of Hochschild (co)homology to relate the three Torelli-type theorems for smooth Fano varieties via a commutative diagram. As an application, we first prove infinitesimal categorical Torelli theorem for a class of prime Fano threefolds. Then we prove a restatement of the Debarre-Iliev-Manivel conjecture infinitesimally., Comment: Correct typos, final version, to appear in Journal of Pure and Applied algebra. arXiv admin note: substantial text overlap with arXiv:2108.02946

We identify the double dual EPW sextic $\widetilde{Y}_{A^{\perp}}$ and the double EPW sextic $\widetilde{Y}_A$, associated with a very general Gushel-Mukai fourfold $X$, with the Bridgeland moduli spaces of stable objects of character $\Lambda_1$ and $\Lambda_2$ in the Kuznetsov component $\mathcal{K}u(X)$. This provides an affirmative answer to a question of Perry-Pertusi-Zhao. As an application, we prove a conjecture of Kuznetsov-Perry for very general Gushel-Mukai fourfolds., Comment: 27 pages, made changes according to referee's suggestions. To appear in Math. Res. Lett. Comments are welcome!

Biomedical entity normalization unifies the language across biomedical experiments and studies, and further enables us to obtain a holistic view of life sciences. Current approaches mainly study the normalization of more standardized entities such as diseases and drugs, while disregarding the more ambiguous but crucial entities such as pathways, functions and cell types, hindering their real-world applications. To achieve biomedical entity normalization on these under-explored entities, we first introduce an expert-curated dataset OBO-syn encompassing 70 different types of entities and 2 million curated entity-synonym pairs. To utilize the unique graph structure in this dataset, we propose GraphPrompt, a prompt-based learning approach that creates prompt templates according to the graphs. GraphPrompt obtained 41.0% and 29.9% improvement on zero-shot and few-shot settings respectively, indicating the effectiveness of these graph-based prompt templates. We envision that our method GraphPrompt and OBO-syn dataset can be broadly applied to graph-based NLP tasks, and serve as the basis for analyzing diverse and accumulating biomedical data., Comment: 12 pages

Understanding protein sequences is vital and urgent for biology, healthcare, and medicine. Labeling approaches are expensive yet time-consuming, while the amount of unlabeled data is increasing quite faster than that of the labeled data due to low-cost, high-throughput sequencing methods. In order to extract knowledge from these unlabeled data, representation learning is of significant value for protein-related tasks and has great potential for helping us learn more about protein functions and structures. The key problem in the protein sequence representation learning is to capture the co-evolutionary information reflected by the inter-residue co-variation in the sequences. Instead of leveraging multiple sequence alignment as is usually done, we propose a novel method to capture this information directly by pre-training via a dedicated language model, i.e., Pairwise Masked Language Model (PMLM). In a conventional masked language model, the masked tokens are modeled by conditioning on the unmasked tokens only, but processed independently to each other. However, our proposed PMLM takes the dependency among masked tokens into consideration, i.e., the probability of a token pair is not equal to the product of the probability of the two tokens. By applying this model, the pre-trained encoder is able to generate a better representation for protein sequences. Our result shows that the proposed method can effectively capture the inter-residue correlations and improves the performance of contact prediction by up to 9% compared to the MLM baseline under the same setting. The proposed model also significantly outperforms the MSA baseline by more than 7% on the TAPE contact prediction benchmark when pre-trained on a subset of the sequence database which the MSA is generated from, revealing the potential of the sequence pre-training method to surpass MSA based methods in general.

We show that a general ordinary Gushel-Mukai(GM) threefold $X$ is reconstructed from the Kuznetsov component $\mathcal{K}u(X)$ together with an extra data coming from tautological sub-bundle of Grassmannian $\mathrm{Gr}(2,5)$. We also prove that $\mathcal{K}u(X)$ determines birational isomorphism class of $X$, while $\mathcal{K}u(X')$ determines the isomorphism class of a general special GM threefold $X'$. As an application, we prove a conjecture of Kuznetsov-Perry in dimension three under a mild assumption. Finally, we use $\mathcal{K}u(X)$ to restate a conjecture of Debarre-Iliev-Manivel regarding fibers of the period map for ordinary GM threefolds., Comment: 39 pages, final version, to appear in Journal of the London Mathematical Society

We study Bridgeland moduli spaces of semistable objects of $(-1)$-classes and $(-4)$-classes in the Kuznetsov components on index one prime Fano threefold $X_{4d+2}$ of degree $4d+2$ and index two prime Fano threefold $Y_d$ of degree $d$ for $d=3,4,5$. For every Serre-invariant stability condition on the Kuznetsov components, we show that the moduli spaces of stable objects of $(-1)$-classes on $X_{4d+2}$ and $Y_d$ are isomorphic. We show that moduli spaces of stable objects of $(-1)$-classes on $X_{14}$ are realized by Fano surface $\mathcal{C}(X)$ of conics, moduli spaces of semistable sheaves $M_X(2,1,6)$ and $M_X(2,-1,6)$ and the correspondent moduli spaces on cubic threefold $Y_3$ are realized by moduli spaces of stable vector bundles $M^b_Y(2,1,2)$ and $M^b_Y(2,-1,2)$. We show that moduli spaces of semistable objects of $(-4)$-classes on $Y_{d}$ are isomorphic to the moduli spaces of instanton sheaves $M^{inst}_Y$ when $d\neq 1,2$, and show that there're open immersions of $M^{inst}_Y$ into moduli spaces of semistable objects of $(-4)$-classes when $d=1,2$. Finally, when $d=3,4,5$ we show that these moduli spaces are all isomorphic to $M^{ss}_X(2,0,4)$., Comment: 30 pages. Some typos are fixed. Add the case $d=3$ in Section 9. Comments are welcome

We study the Hilbert scheme $\mathcal{H}$ of twisted cubics on a special smooth Gushel-Mukai threefolds $X_{10}$. We show that it is a smooth irreducible projective threefold if $X_{10}$ is general among special Gushel-Mukai threefolds, while it is singular if $X_{10}$ is not general. We construct an irreducible component of a moduli space of Bridgeland stable objects in the Kuznetsov component of $X_{10}$ as a divisorial contraction of $\mathcal{H}$. We also identify the minimal model of Fano surface $\mathcal{C}(X_{10}')$ of conics on a smooth ordinary Gushel-Mukai threefold with moduli space of Bridgeland stable objects in the Kuznetsov component of $X_{10}'$. As a result, we show that the Kuznetsov's Fano threefold conjecture is not true, Comment: 35 pages, substantially rewritten, remove everything irrelevant. Fix typos and correct mistakes on computations, prove stronger results via Serre-invariant stability conditions

We study exceptional collections of line bundles on surfaces. We prove that any full cyclic strong exceptional collection of line bundles on a rational surface is an augmentation in the sense of L.Hille and M.Perling. We find simple geometric criteria of exceptionality (strong exceptionality, cyclic strong exceptionality) for collections of line bundles on weak del Pezzo surfaces. As a result, we classify smooth projective surfaces admitting a full cyclic strong exceptional collection of line bundles. Also, we provide an example of a weak del Pezzo surface of degree 2 and a full strong exceptional collection of line bundles on it which does not come from augmentations, thus answering a question by Hille and Perling., Comment: This is a seriously rewritten part of the paper arXiv.math:1710.03972, accepted to EJM. Here we concentrate on the cyclic strong exceptional collections. 35 pages

Let $X$ be a del Pezzo surface. When the degree of $X$ is at least 4, we compute the cohomology of a general sheaf in the moduli space of Gieseker semistable sheaves. We also classify the Chern characters for which the general sheaf in the moduli space is non-special, i.e. has at most one nonzero cohomology group. Our results hold for arbitrary polarizations, slope semistability, and semi-exceptional moduli spaces. When the degree of $X$ is at least 3, we further show our construction of certain vector bundles implies the existence of stable and semistable sheaves with respect to the anti-canonical polarization., Comment: update the email address of the second author, update the references, correct typos. To appear in Annales de Institut Fourier

Suppose $S$ is a smooth projective surface over an algebraically closed field $k$, $\mathcal{L}=\{L_1,\ldots,L_n\}$ is a full strong exceptional collection of line bundles on $S$. Let $Q$ be the quiver associated to this collection. One might hope that $S$ is the moduli space of representations of $Q$ with dimension vector $(1,\ldots,1)$ for a suitably chosen stability condition $\theta$: $S\cong M_\theta$. In this paper, we show that this is the case for del Pezzo surfaces. Furthermore, we show the blow-up at a point can be recovered from an augmentation of exceptional collections (in the sense of L. Hille and M.Perling) via morphism between moduli of quiver representations., Comment: 34 pages. New version deals only with exceptional collections on del Pezzo surfaces with cleaner presentation. For general cases, please see the old version

In this note we apply the techniques of the toric systems introduced by Hille-Perling to several problems on smooth projective surfaces: We showed that the existence of full exceptional collection of line bundles implies the rationality for small Picard rank surfaces; we proved equivalences of several notions of cyclic strong exceptional collection of line bundles; we also proposed a partial solution to a conjecture on exceptional sheaves on weak del Pezzo surfaces., Comment: 24 pages, preliminary sections 2-5 are overlap with the earlier work arXiv:1710.03972. Section 7.1 added: improved a theorem on characterization of quadrics by full exceptional collection via pseudo anti-canonical height. Comments are welcome

We study full exceptional collections of line bundles on surfaces. We prove that any full strong exceptional collection of line bundles on a weak del Pezzo surface of degree $\ge 3$ is an augmentation in the sense of L.Hille and M.Perling, while for some weak del Pezzo surfaces of degree $2$ the above is not true. We classify smooth projective surfaces possessing a cyclic strong exceptional collection of line bundles of maximal length: we prove that they are weak del Pezzo surfaces and find all types of weak del Pezzo surfaces admitting such a collection. We find simple criteria of exceptionality/strong exceptionality for collections of line bundles on weak del Pezzo surfaces., Comment: 61 pages, files with computer programs are attached. Comments are welcome