By Massey

This ebook is meant to function a textbook for a path in algebraic topology at the start graduate point. the most subject matters coated are the type of compact 2-manifolds, the elemental workforce, masking areas, singular homology concept, and singular cohomology idea. those issues are built systematically, averting all unecessary definitions, terminology, and technical equipment. anyplace attainable, the geometric motivation at the back of a number of the techniques is emphasised. The textual content involves fabric from the 1st 5 chapters of the author's previous publication, ALGEBRAIC TOPOLOGY: AN advent (GTM 56), including just about all of the now out-of- print SINGULAR HOMOLOGY conception (GTM 70). the cloth from the sooner books has been rigorously revised, corrected, and taken brand new.

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**Sample text**

8]). We introduce two rational numbers measuring how far is the multidegree d from being balanced. d / 0. d ; W / Ä 0 for any subcurve W Â X . d / D 0. 9) 30 3 Combinatorial Results Let us prove the second additive formula; the proof of the first one is similar and left to the reader. 11) We will prove this for SdC ; the proof for Sd works exactly in the same way. W1 \ W2 /. d / D W1 \ W2 2 SdC . e. d /c 2 Sd . d / do not have common irreducible components. d /. d /. Let us show first how, using the claim, we can conclude the proof of the Lemma.

Proof The proof is a generalization of [Cap94, Prop. 3]. Let us first prove the implication (i))(ii). d; g d 6Á g 1/ D 1; (*) 1 mod 2: (**) 38 3 Combinatorial Results Let X be a quasi-wp-stable curve of genus g and let L be any properly balanced line bundle on X of degree d . Call d the multidegree of L. 25) is an exceptional component of X . X /. X / D 0; g 1; 2g 2. 10) and we are done. X / D 2g 2 which implies that Z c is an exceptional component. 25). 25) by g 1 and taking congruence modulo 2, we obtain that d Á kZ mod 2.

Y/ ! S is a family of p-stable curves of genus g, called the p-stable reduction of v W Y ! S . For every geometric point s 2 S , the morphism s W Ys ! Y/s . Moreover, the formation of the p-stable reduction commutes with base change. wp p This defines a morphism of stacks ps W Mg ! Mg . Proof If v W Y ! S is a family of stable curves, the statement was proved by Hassett-Hyeon in [HH09, Sect. 3] under the assumption that g 3 and then extended to g D 2 with a similar argument by Heyon-Lee in [HL07, Sect.