# Professor Paolo Cascini

Faculty of Natural SciencesDepartment of Mathematics

Professor of Mathematics

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### Contact

+44 (0)20 7594 8861p.cascini

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### Location

676Huxley BuildingSouth Kensington Campus

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## Publications

Publication Type
Year
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35 results found

Cascini P, Martinez-Garcia J, Rubinstein YA, 2022, On the body of ample angles of asymptotically log Fano varieties, Rendiconti del Circolo Matematico di Palermo Series 2, ISSN: 0009-725X

In dimension two, we reduce the classification problem for asymptotically log Fano pairs to the problem of determining generality conditions on certain blow-ups. In any dimension, we prove the rationality of the body of ample angles of an asymptotically log Fano pair, i.e., these convex bodies are always rational polytopes.

Journal article

Ambro F, Cascini P, Shokurov V, Spicer Cet al., 2021, Positivity of the Moduli Part

We prove the Cone Theorem for algebraically integrable foliations. As aconsequence, we show that termination of flips implies the b-nefness of themoduli part of a log canonical pair with respect to a contraction, generalisingthe case of lc trivial fibrations.

Journal article

Cascini P, 2021, New directions in the Minimal Model Program, Bollettino Della Unione Matematica Italiana, Vol: 14, Pages: 179-190, ISSN: 0041-7084

We survey some recents developments in the Minimal Model Program. After an elementary introduction to the program, we focus on its generalisations to the category of foliated varieties and the category of varieties defined over any algebraically closed field of positive characteristic.

Journal article

Cascini P, Spicer C, 2021, On the MMP for rank one foliations on threefolds, Inventiones Mathematicae, Vol: 225, Pages: 603-690, ISSN: 0020-9910

We prove existence of flips and the base point free theorem for log canonicalfoliated pairs of rank one on a Q-factorial projective klt threefold. This, inparticular, provides a proof of the existence of a minimal model for a rank onefoliation on a threefold for a wider range of singularities, after McQuillan.Moreover, we show abundance in the case of numerically trivial log canonicalfoliated pairs of rank one.

Journal article

Cascini P, Ejiri S, Kollar J, Zhang Let al., 2021, Subadditivity of Kodaira dimension does not hold in positive characteristic, Commentarii Mathematici Helvetici, Vol: 96, Pages: 465-481, ISSN: 0010-2571

Over any algebraically closed field of positive characteristic, we con-struct examples of fibrations violating subadditivity of Kodaira dimension.

Journal article

Cascini P, Meng S, Zhang D-Q, 2020, Polarized endomorphisms of normal projective threefolds in arbitrarycharacteristic, Mathematische Annalen, Vol: 378, Pages: 637-665, ISSN: 0025-5831

Let X be a projective variety over an algebraically closed field k of arbitrary characteristic p≥0. A surjective endomorphism f of X is q-polarized if f∗H∼qH for some ample Cartier divisor H and integer q>1. Suppose f is separable and X is Q-Gorenstein and normal. We show that the anti-canonical divisor −KX is numerically equivalent to an effective Q-Cartier divisor, strengthening slightly the conclusion of Boucksom, de Fernex and Favre (Duke Math J 161(8):1455–1520, 2012, Theorem C) and also covering singular varieties over an algebraically closed field of arbitrary characteristic. Suppose f is separable and X is normal. We show that the Albanese morphism of X is an algebraic fibre space and f induces polarized endomorphisms on the Albanese and also the Picard variety of X, and KX being pseudo-effective and Q-Cartier means being a torsion Q-divisor. Let fGal:X¯¯¯¯→X be the Galois closure of f. We show that if p>5 and co-prime to degfGal then one can run the minimal model program (MMP) f-equivariantly, after replacing f by a positive power, for a mildly singular threefold X and reach a variety Y with torsion canonical divisor (and also with Y being a quasi-étale quotient of an abelian variety when dim(Y)≤2). Along the way, we show that a power of f acts as a scalar multiplication on the Neron-Severi group of X (modulo torsion) when X is a smooth and rationally chain connected projective variety of dimension at most three.

Journal article

Cascini P, Tanaka H, 2020, Relative semi-ampleness in positive characteristic, Proceedings of the London Mathematical Society, Vol: 121, Pages: 617-655, ISSN: 0024-6115

Given an invertible sheaf on a fibre space between projective varieties ofpositive characteristic, we show that fibrewise semi-ampleness implies relativesemi-ampleness. The same statement fails in characteristic zero.

Journal article

Cascini P, Tanaka H, 2019, Purely log terminal threefolds with non-normal centres in characteristic two, American Journal of Mathematics, Vol: 141, Pages: 941-979, ISSN: 0002-9327

We show that many classical results of the minimal model programme do not hold over an algebraically closed field of characteristic two. Indeed, we construct a three dimensional plt pair whose codimension one part is notnormal, a three dimensional klt singularity which is not rational nor Cohen-Macaulay, and a klt Fano threefold with non-trivial intermediatecohomology.

Journal article

Cascini P, Floris E, 2018, On invariance of plurigenera for foliations on surfaces, Journal für die reine und angewandte Mathematik (Crelles Journal), Vol: 2018, Pages: 201-236, ISSN: 1435-5345

We show that if $(X_t,\mathcal{F}_t)_t$ is a family of foliations withreduced singularities on a smooth family of surfaces, then invariance ofplurigenera holds for sufficiently large $m$. On the other hand, we provideexamples on which the result fails, for small values of $m$.

Journal article

Cascini P, Spicer C, 2018, MMP for co-rank one foliation on threefolds, Publisher: arXiv

We prove existence of flips, special termination, the base point free theoremand, in the case of log general type, the existence of minimal models for F-dltfoliated log pairs of co-rank one on a projective threefold. As applications, we show the existence of F-dlt modifications andF-terminalisations for foliated log pairs and we show that foliations withcanonical or F-dlt singularities admit non-dicritical singularities. Finally,we show abundance in the case of numerically trivial foliated log pairs.

Working paper

Cascini P, Tanaka H, 2018, Smooth rational surfaces violating Kawamata–Viehweg vanishing, European Journal of Mathematics, Vol: 4, Pages: 162-176, ISSN: 2199-675X

We show that over any algebraically closed field of positive characteristic, there exists a smooth rational surface which violates Kawamata–Viehweg vanishing.

Journal article

Cascini P, Tanaka H, Witaszek J, 2017, On log del Pezzo surfaces in large characteristic, Compositio Mathematica, Vol: 153, Pages: 820-850, ISSN: 0010-437X

We show that any Kawamata log terminal del Pezzo surface over an algebraically closed field of large characteristic is globally F -regular or it admits a log resolution which lifts to characteristic zero. As a consequence, we prove the Kawamata–Viehweg vanishing theorem for klt del Pezzo surfaces of large characteristic.

Journal article

Cascini P, Tasin L, 2017, On the Chern numbers of a smooth threefold, Transactions of the American Mathematical Society, ISSN: 0002-9947

We study the behaviour of Chern numbers of three dimensional terminalvarieties under divisorial contractions.

Journal article

Cascini P, Tanaka H, Witaszek J, 2017, Klt del Pezzo surfaces which are not Globally F-split, International Mathematics Research Notices, Pages: rnw300-rnw300, ISSN: 1073-7928

Journal article

Bisi C, Cascini P, Tasin L, 2016, A Remark on the Ueno-Campana's Threefold, Michigan Mathematical Journal, Vol: 65, Pages: 567-572, ISSN: 0026-2285

We show that the Ueno–Campana’s threefold cannot beobtained as the blow-up of any smooth threefold along a smooth center,answering negatively a question raised by Oguiso and Truong.

Journal article

Cascini P, McKernan J, Pereira JV, 2016, Foliation Theory in Algebraic Geometry, Publisher: Springer, ISBN: 9783319244600

Featuring a blend of original research papers and comprehensive surveys from an international team of leading researchers in the thriving fields of foliation theory, holomorphic foliations, and birational geometry, this book presents the ...

Book

Cascini P, Gongyo Y, Schwede K, 2015, Uniform bounds for strongly -regular surfaces, Transactions of the American Mathematical Society, Vol: 368, Pages: 5547-5563, ISSN: 1088-6850

We show that if (X, B) is a two dimensional Kawamata logterminal pair defined over an algebraically closed field of characteristicp, and p is sufficiently large, depending only on the coefficients of B,then (X, B) is also strongly F-regular.

Journal article

Cascini P, Tanaka H, Xu C, 2015, On base point freeness in positive characteristic, Annales scientifiques de l'École normale supérieure, Vol: 48, Pages: 1239-1272, ISSN: 0012-9593

We prove that if (X, A + B) is a pair defined over an algebraicallyclosed field of positive characteristic such that (X, B) is strongly F-regular,A is ample and KX + A + B is strictly nef, then KX + A + B is ample.Similarly, we prove that for a log pair (X, A + B) with A being ample andB effective, KX + A + B is big if it is nef and of maximal nef dimension.As an application, we establish a rationality theorem for the nef thresholdand various results towards the minimal model program in dimension three inpositive characteristic.

Journal article

Cascini P, Lazic V, 2014, On the number of minimal models of a log smooth threefold, JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES, Vol: 102, Pages: 597-616, ISSN: 0021-7824

Journal article

Cascini P, 2013, Rational Curves on Complex Manifolds, MILAN JOURNAL OF MATHEMATICS, Vol: 81, Pages: 291-315, ISSN: 1424-9286

Journal article

Cascini P, Hacon C, Mustata M, Schwede Ket al., 2012, On the numerical dimension of pseudo-effective divisors in positive characteristic, American Journal of Math. (to appear)

Let X be a smooth projective variety over an algebraically closed field ofpositive characteristic. We prove that if D is a pseudo-effective R-divisor onX which is not numerically equivalent to the negative part in its divisorialZariski decomposition, then the numerical dimension of D is positive. Incharacteristic zero, this was proved by Nakayama using vanishing theorems.

Journal article

Cascini P, Nakamaye M, 2012, Seshadri constant on smooth threefolds, Advances in Geometry (to appear)

We prove that the Seshadri constant of an ample line bundle at a very general point of a smooth projective threefold is larger than 1/2. While falling short of the conjectured lower bound of one, this improves on known results. We systematically exploit new results and ideas related to the variation and complexity of base loci.

Journal article

Cascini P, Lazic V, 2012, New outlook on the Minimal Model Program, I, Duke Mathematical Journal, Vol: 161, Pages: 2415-2467

We give a new and self-contained proof of the finite generation of adjoint rings with big boundaries. As a consequence, we show that the canonical ring of a smooth projective variety is finitely generated.

Journal article

Cascini P, Zhang D-Q, 2012, Effective finite generation for adjoint rings, Annales de l'institut Fourier (to appear)

We describe a bound on the degree of the generators for some adjoint rings on surfaces and threefolds.

Journal article

Cascini P, McKernan J, Mustata M, 2012, The augmented base locus in positive characteristic, Proceedings of the Edinburgh Mathematical Society

Journal article

Cascini P, Panov D, 2012, Symplectic generic complex structures on 4-manifolds with b+ = 1, Journal of Symplectic Geometry, Vol: 10, Pages: 1-10

Journal article

Cascini P, Lazic V, 2011, The Minimal Model Program revisited, Contributions to Algebraic Geometry

We give a light introduction to some recent developments in Mori theory, and to our recent direct proof of the nite generation of the canonical ring.

Journal article

Birkar C, Cascini P, Hacon C, Mckernan Jet al., 2010, Existence of Minimal Models for Varieties of LogGeneral Type, J. Amer. Math. Soc.

Journal article

P Cascini, 2006, Subsheaves of the Cotangent Bundle, Central European Journal of Math., Vol: 4, Pages: 209-224

For any smooth projective variety, we study a birational invariant, deﬁned by Campana which depends on the Kodaira dimension of the subsheaves of the cotangent bundle of the variety and its exterior powers. We provide new bounds for a related invariant in any dimension and in particular we show that it is equal to the Kodaira dimension of the variety, in dimension up to 4, if this is not negative.

Journal article

FA Bogomolov, PCascini, B de Oliveira, 2006, Singularities on Complete Algebraic Varieties, Central European Journal of Math, Vol: 4, Pages: 194-208

We prove that any finite set of n-dimensional isolated algebraic singularities can be afforded on a simply connected projective variety.

Journal article

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