Albanese maps and fundamental groups of varieties with many rational points over function fields. Specifically, we present a surprisingly elementary and concrete proof of the asymptotic nonvanishing of veronese syzygies, and we obtain effective results for arithmetically cohenmacaulay varieties. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros the fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of. We study the asymptotic behavior of the syzygies of a smooth projective variety as the positivity of the embedding line bundle grows. It is made up mainly from the material in referativnyi zhurnal matematika during 19651973 and is devoted to the geometric aspects of the theory of algebraic varieties. We study the asymptotic behavior of veronese syzygies as representations of the general linear group.
We prove that as least as far as grading is concerned, the minimal resolution of the ideal of x has a surprisingly uniform asymptotic shape. Asymptotic syzygies of algebraic varieties by lawrence ein and robert lazarsfeld get pdf 399 kb. On the cohomologies of cohen macaulay determinantal subvarieties of pn geometry of complex projective varieties, seminar and conferences, vol. This provides the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semiample fashion behave in nuanced and previously unseen ways. Algebraic geometry and statistical learning theory by. Projective normality and syzygies of algebraic surfaces. C, and their subspaces known as algebraic varieties.
The goal of this note is to quantitatively study the behavior of asymptotic syzygies for certain toric surfaces, including hirzebruch surfaces. Effective nonvanishing of asymptotic syzygies effective nonvanishing of asymptotic syzygies. Introduction the purpose of this thesis is the exposition of some recent results about syzygies of projective varieties. Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Ams proceedings of the american mathematical society.
Ein vanishing theorems, a theorem of severi, and the equations. Ein syzygies and koszul cohomology of smooth projective varieties of arbitrary dimension, invent. Semiample asymptotic syzygies algebraic geometry and commutative algebra seminar i will discuss the asymptotic nonvanishing of syzygies for products of projective spaces, generalizing the monomial methods of einermanlazarsfeld. Ill present joint work with lawrence ein concerning the asymptotic behavior of the syzygies of a smooth projective variety x as the positivity of the embedding line bundle grows. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. The main result asserts that the syzygy modules are nonzero in almost all degrees allowed by castelnuovomumford regularity. American mathematical society 201 charles street providence, rhode island 0290422 4014554000 or 8003214267 ams, american mathematical society, the tricolored ams logo, and advancing research, creating connections, are trademarks and services marks of the american mathematical society and registered in the u. In the special case of a segre variety this was conjectured by garcia, stillman and sturmfels, inspired by work on algebraic statistics, as well as by pachter and sturmfels, inspired by work on phylogenetic inference. D syzygies of algebraic varieties 115 notes 119 2 linear series 121 2. Syzygies and the structure of free resolutions are currently the subject of intense research, though i have mentioned only a few highlights. I will discuss the implications of the proof of the effective statement for toric. In mathematics, hilberts syzygy theorem is one of the three fundamental theorems about polynomial rings over fields, first proved by david hilbert in 1890, which were introduced for solving important open questions in invariant theory, and are at the basis of modern algebraic geometry. Projective normality and syzygies of algebraic surfaces dedicated to david eisenbud on his fiftieth birthday by f. This paper studies the asymptotic behavior of the syzygies of a smooth projective variety x as the positivity of the embedding line bundle grows.
Introduction to syzygies linear algebra studies the properties of vector spaces and their maps. Recently, i have become particularly interested in the study of syzygies under various asymptotic constructions, and in the study of algebraic geometry over finite fields. Nov 12, 2014 we study the asymptotic behavior of veronese syzygies as representations of the general linear group. The purpose of this thesis is the exposition of some recent results about syzygies of projective varieties.
The purpose of the paper is to give a sharp asymptotic description of the weights that appear in the syzygies of a smooth toric variety. This provides the first example of how the asymptotic syzygies of a smooth projective variety whose. Let l be a very ample line bundle on x inducing a projectively normal embedding x. Ullery on a generalization of gonality to higher dimensional varieties and show how these invariants can be used as a measure of the irrationality of an algebraic variety. We prove that the ideal of the variety of secant lines to a segreveronese variety is generated in degree three by minors of flattenings. It will be an important tool to analyze the topology of x. Greens seminal paper 5 that popularized the notion that the syzygies of x can be viewed as a type of cohomology, called koszul cohomology, attached to the pair. Effective nonvanishing of asymptotic adjoint syzygies. The first part presents the basic definitions and results relative to to the language of graded betti numbers and koszul cohomology. Topology of algebraic varieties universiteit utrecht. A quick proof of nonvanishing for asymptotic syzygies. Asymptotic syzygies of algebraic varieties springerlink. We prove that as the positivity of the embedding increases, in any strand of syzygies, torus weights after.
Effective nonvanishing of asymptotic syzygies math. Although we will look at some vanishing results, our focus will primarily be on proving nonvanishing of syzygies through an induction involving secant constructions, which is an approach developed by lazarsfeld and ein in their paper asymptotic syzygies of algebraic varieties. Classically, it is the study of the zero sets of polynomials. Asymptotic syzygies in the setting of semiample growth. Pdf a quick proof of nonvanishing for asymptotic syzygies. Asymptotic syzygies and higher order embeddings, arxiv. Title syzygies of algebraic varieties we begin with some. Recent work has indicated that subjects such as fultonmcpherson intersection theory, the hilbert scheme of points, and the kempfweyman method for computing syzygies all have something to add to complexity theory. Asymptotic syzygies of normal crossing varieties core. We begin with some classical results of syzygies of algebraic curves and introduction to koszul cohomolgy of mark green. If x is a projective scheme and l a line bundle on x, we will write l. In this thesis we study asymptotic syzygies of algebraic varieties and equations of abelian surfaces, with applications to cyclic covers of genus two curves.
I will discuss recent work attempting to generalize these conjectures to the case when the. I will discuss the asymptotic nonvanishing of syzygies for products of projective spaces, generalizing the monomial methods of einermanlazarsfeld. Ams transactions of the american mathematical society. Algebra seminar march 2018 asymptotic syzygies in the semiample setting university of wisconsin algebra and algebraic geometry seminar february 2018 asymptotic syzygies in the semiample setting poster san diego, ca joint math meetings awm poster session january 2018 asymptotic syzygies in the semiample setting university of. The two other theorems are hilberts basis theorem that asserts that all ideals of polynomial rings over a. Syzygies and koszul cohomology of smooth projective varieties of arbitrary dimension with r. For an account of the use of syzygies in algebraic geometry. We also give an effective statement for veronese varieties that we conjecture to be optimal. Pdf download a royal road to algebraic geometry download. Review of the birational geometry of curves and surfaces the minimal model program for 3folds towards the minimal model program in higher dimensions the strategy the conjectures of the mmp mild singularities. Pdf download a royal road to algebraic geometry download full ebook. On triangular decompositions of algebraic varieties m. Msri representation theory, homological algebra, and free.
Asymptotic aspects of complex and algebraic geometry june. For geometers, i expect that complexity theory will be as good a source for questions in algebraic geometry as has been modern physics. Moreno maza1 computational mathematics group, nag ltd, oxford ox2 8dr, greatbritain abstract di. While interest in the syzygies of a projective variety x goes back to hilbert, it was mark l. We give a simple proof of voisins theorem for general canonical curves of even genus. It is made up mainly from the material in referativnyi zhurnal matematika during 19651973. Pdf asymptotic behaviour of numerical invariants of. Algebraic geometry and number theory seminar september 2018 syzygies of surfaces via distributed computing depaul university algebra seminar march 2018 asymptotic syzygies in the semiample setting university of wisconsin algebra and algebraic geometry seminar february 2018 asymptotic syzygies in the semiample setting poster san. Finally, we propose a number of questions and open problems concerning asymptotic syzygies of higherdimensional varieties. Schur asymptotics of veronese syzygies springerlink. The purpose of the paper is to give a sharp asymptotic description of the weights that appear in the syzygies of a toric variety. Asymptotic syzygies of algebraic varieties article pdf available in inventiones mathematicae 1903 march 2011 with 37 reads how we measure reads.
Asymptotic syzygies of algebraic varieties lazarsfeld, robert aug 16, 2012 page metadata. In particular, we show that the asymptotic linear syzygies of hirzebruch surfaces embedded by \\mathcalod,2\ conform to. Ill also discuss my joint work with rob lazarsfeld on asymptotic syzygies, and the gonality conjecture. Christopher hacon the birational geometry of algebraic varieties. Banff international research station for mathematical innovation and discovery.
All these ideas have brought into light many asymptotic tools, and uncovered new insights of analytic nature like approximation of metrics, of geometric nature like positivity aspects and vanishing theorems, and of algebraic nature like the study of syzygies of an algebraic variety. While interest in the syzygies of a projective variety x goes back to hilbert, it was. We study the asymptotic nonvanishing of syzygies for products of projective spaces. The idea is that one can reduce the statements to some simple computations with monomials. People algebraic geometry, applied algebra, and number. Pages in category algebraic varieties the following 55 pages are in this category, out of 55 total. Ideals, varieties, and algorithms is a book where you learn by doing. For veronese embeddings of projective space, we give an effective statement that in some cases is optimal, and conjecturally always is so. We prove that as the positivity of the embedding increases, in any strand of syzygies, torus weights after normalization stabilize to. Higher dimensional algebraic geometry in honour of professor. Recall that a line bundle b on a smooth projective variety x is said to be pjet. Workshop on syzygies in algebraic geometry, with an. This shows that asymptotically veronese syzygies have a very rich.
If time allows, i will also briefly describe asymptotic syzygies of higher dimensional algebraic varieties. Asymptotic aspects of complex and algebraic geometry under the auspices of the phd program in mathematics at university of milano bicocca. The purpose of this thesis is to expose some results relative to the asymptotic behaviour of the graded betti numbers of algebraic varieties, as the positivity of the embedding grows. Sections renumbered to conform to published version.
Pdf a simple proof of voisins theorem for canonical. Asymptotic syzygies of algebraic varieties stony brook mathematics. Algebraic geometrycommutative algebra seminar, department. Syzygies of projective varieties of large degree semantic scholar. Pdf a simple proof of voisins theorem for canonical curves. Lawrence man hou ein born 18 november 1955 is a mathematician who works in algebraic geometry education and career. The dotted curves in the lower gure are the asymptotic approximations for the roots close to 1. Nevertheless, there is a growing body of results relating fundamental properties in algebraic geometry and commutative algebra to the structure of equations. Asymptotic syzygies of a normal crossing variety follow the same vanishing behavior as one of its smooth components, unless there is a cohomological obstruction arising from how the smooth components intersect each other. Lawrence ein received in 1976 his bachelors degree from ucla and in 1981 his ph. Turning to details, let x be a smooth projective variety of dimension n defined over an. The treatment is linear, and many simple statements are left for the reader to prove as exercises. For a fixed homological degree p of the syzygies, we describe the exact asymptotic growth for the number of distinct irreducible representations and for the number of irreducible representations also counting multiplicities. Invited talks at workshops and conferences september 7 11, 2020.
Asymptotic weights of syzygies of toric varieties sciencedirect. A second course in algebraic geometry and commutative algebra. Effective nonvanishing of asymptotic syzygies april 30, 20 04. Asymptotic syzygies of algebraic varieties tesi di laurea magistrale relatore. I am interested in the study of syzygies and in the use of free complexes in algebraic geometry and commutative algebra. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. The second part is dedicated to the recent results of l. We prove that at least as far as grading is concerned, the minimal resolution of the ideal of x has a.
Read the geometry of syzygies a second course in algebraic geometry and commutative algebra pdf online. Daniel erman algebraic geometry, applied algebra, and. Affineness of the complement of the ramification locus. In recent years numerous conjectures have been made describing the asymptotic betti numbers of a projective variety as the embedding line bundle becomes more ample. The purpose of this paper is to establish an effective nonvanishing theorem for the syzygies of an adjointtype line bundle on a smooth variety, as the positivity of the embedding increases. To rst approximation, a projective variety is the locus of zeroes of a system of homogeneous polynomials. For curves, a conjecture of green and lazarsfeld would predict essentially that the shape of the minimal resolution of a very high degree embedding is controlled by the gonality of the curve. Asymptotic syzygies of algebraic varieties ubc library. In this work we develop new techniques to compute koszul cohomology groups for several classes of varieties. Sure to be influential, this book lays the foundations for the use of algebraic geometry in statistical learning theory.
We will also use various sources for commutative algebra. Asymptotic weights of syzygies of toric varieties core. This completely determines the terms of the minimal free resolution of the coordinate ring of such curves. We prove that as the positivity of the embedding increases, in any strand of syzygies, torus weights after normalization stabilize to the same fixed shape that we explicitly specify. Many widely used statistical models and learning machines applied to information science have a parameter space that is singular. Asymptotic syzygies of algebraic varieties 5 and eisenbud et.
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